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Full text of "Prediction of cohesive sediment movement in estuarial waters"

ACKNOWLEDGEMENTS 



The author would like to express his sincerest appreciation to his 
research advisor and supervisory committee co-chairman. Or. A.J. Mehta, 
Associate Professor of Coastal and Oceanographic Engineering, for his 
continuous guidance and encouragement throughout the course of this 
research. Appreciation is also extended for the valuable advise and 
suggestions of the supervisory committee chairman. Dr. B.A. Christensen, 
Professor of Civil Engineering, as well as for the guidance received 
from the other committee members: Dr. B.A. Benedict, Professor of Civil 
Engineering; and Dr. J.L. Eades, Associate Professor of Geology. 
Sincere thanks are also due to Dr. E. Partheniades, Professor of 
Engineering Sciences, whose interest and suggestions were of a great 
help to the author. 

The author wishes to acknowledge the assistance provided by Dr. 
Ranjan Ariathurai to this investigation. He provided a copy of the 
finite element solution routine used in this study and gave advice 
regarding its use and related experimental research. 

Appreciation is extended to Drs. D.G. Bloomquist and P. Nielsen for 
their suggestions and providing references on various aspects of this 
study. 

Special thanks go to Mr. Vernon Sparkman and the staff of the 
Coastal Engineering Laboratory for their assistance with the experiments 

ii 



PREDICTION OF COHESIVE SEDIMENT MOyEKEMT 
IN ESTUARIAL WATERS 



By 



EARL JOSEPH HAYTER 



DISSERTATION PRESENTED TO THE GRADUATE COUNCIL 
OF THE UNIVElSTTf OF PWRIDA IN 
PARTIAL FULFILLMENT OF THE REOUIREMENTS 
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 
1983 



performed during this research. The author wishes to thank Ms. Debbie 
LaMar for typing this manuscript and Ms. Lillean Pieter for drafting the 
figures. In addition, the author thanks Ms. Lucile Lehmann and Ms. 
Helen Twedell of the Coastal Engineering Archives for their assistance. 

Great appreciation is extended to the Water Resources Division of 
the U.S. Geological Survey for their financial support of this research 
through the thesis support program. More specifically, the author 
expresses his gratitude to Dr. Robert Baker, regional research 
hydrologist, and to Drs. Carl Goodwin and Harvey Jobson who served as 
thesis support advisors. 

Finally, the author thanks his wife Janet for her love, moral 
encouragement and patience, and his parents, George and Lois Hayter, for 
their love and support. 



iii 



TABLE OF CONTENTS 

PAGE 

ACKNOWLEDGEMENTS ii 

LIST OF TABLES viii 

LIST OF FIGURES ix 

ABSTRACT xvii 

CHAPTER 

I INTRODUCTION . 1 

1.1. Estuarial Cohesive Sediment 

Dynamics 1 

1.2. Sediment Related Problems 

in Estuaries 4 

1.3. Approach to the Problems 7 

1.4. Scope of Investigation 9 

II BACKGROUND MATERIAL 12 

2.1. Introductory Note 12 

2.2. Description and Properties 

of Cohesive Sediments 12 

2.2.1. Composition 12 

2.2.2. Origin 13 

2.2.3. Structure 14 

2.2.4. Interparti cle Forces 15 

2.2.5. Cation Exchange Capacity .... 19 

2.2.6. Coagulation 21 

2.3. Significance of Important Physical 
Factors in Estuarial Sediment 

Transport 30 

2.3.1. Estuarial Dynamics 30 

2.3.2. Sediment Processes 35 

III SEDIMENT TRANSPORT MECHANICS 47 

3.1. Introductory Note 47 

i V 



PAGE 



3.2. Governing Equations 47 

3.2.1. Coordinate System 47 

3.2.2. Equations of Motion 47 

3. 2.1. a. Continuity , 49 

3.2.1.b. Conservation of 

Momentum 49 

3.2.3. Advection-Dispersion 

Equation 51 

3.3. Sediment Bed 53 

3.3.1. Bed Structure 53 

3.3.2. Effect of Salinity on 

Bed Structure 64 

3.3.3. Bed Schematization 71 

3.4. Erosion 75 

3.4.1. Previous Investigations 75 

3.4.2. Effect of Salinity 

on Resuspension 92 

3.4.3. Erosion Algorithm 101 

3.5. Dispersive Transport 106 

3.5.1. Dispersion Mechanisms 106 

3.5.2. Dispersion Algorithm 112 

3.6. Deposition 116 

3.6.1. Previous Investigations 116 

3.6.2. Effect of Salinity 

on Deposition 133 

3.6.3. Deposition Rates 140 

3.6.4. Deposition Algorithm 152 

3.7. Consolidation 154 

3.7.1. Description 154 

3.7.2. Consolidation Algorithm 168 

IV MODEL DEVELOPMENT 178 

4.1. Introductory Note 178 

4.2. Review of Previous Models •^^^ 

4.3. Model Description 181 

4.4. Finite Element Formulation 188 

4.4.1. Introductory Note 188 

4.4.2. Shape Functions 189 

4.4.3. Galerkin Weighted 

Residual Method 195 

4.4.4. Equation Solvers 201 

4.5. Convergence and Stability 201 

V 



PAGE 



V MODEL VERIFICATION AND APPLICATION 203 

5.1. Introductory Note 203 

5.2. Laboratory Experiments 203 

5.2.1. Recirculating Flume 

Experiments 204 

5. 2.1. a. Facilities 204 

5.2. l.b. Instrumentation. . . . 207 

5. 2. I.e. Procedure 217 

5.2.1. d. Results 222 

5.2.2. Rotating Annular Flume 

Experiment 227 

5. 2. 2. a. Facilities 227 

5.2.2.b. Instrumentation. . . , 230 

5.2.2.C. Procedure 230 

5.2.2.d. Results 233 

5.2.3. Model Simulations 233 

5.3. Simulation of WES Deposition 

Experiment 236 

5.4. Discussion of Results , 238 

5.5. Model Applications 246 

5.6. Model Limitations 257 

5.7. Model Applicability , 258 

5.7.1. Water Quality Problems ..... 258 

5.7.2. Sedimentation Management 

Problems 259 

VI CONCLUSIONS AND RECOMMENDATIONS 260 

6.1. Summary and Conclusions 260 

6.2. Recommendations for Future 

Research 269 

APPENDICES 

A Derivation of Advection-Di spersi on 

Equation 271 

B Coefficient Matrices in the Element 

Matrix Differential Equation 286 

vi 



PAGE 



C Computer Program . 289 

C.l. Main Program 289 

C.2. Subroutines 290 

C.3. Flow Chart 296 

C. 4. User's Manual „ 308 

0 Data Collection and Analysis 

Programs 323 

D. l. Field Data Collection 

Program , 323 

D.2. Laboratory Sediment Testing 

Program 330 

REFERENCES 335 

BIOGRAPHICAL SKETCH 349 



vi i 



LIST OF TABLES 

TABLE PAGE 

2.1 Properties of Sediment Aggregates (after Krone, 

1963) 41 

3.1 Principle Factors Controlling Erosion of 

Saturated Cohesive Sediment Beds 77 

3.2 Cation Concentrations in Processed Sodium 

Chloride and Standard Sea Salt. 95 

3.3 Variation of Empirical Coefficients in the 
Relationship Between Pi^^^) and T^^ 171 

4.1 Quadratic Shape Functions 192 

4.2 Derivatives of Shape Functions 194 

5.1 Chemical Composition of the Tap Water 

(after Dixit, 1982) 217 



V i i i 



LIST OF FIGURES 



FIGURE PAGE 

1.1 Schematic Representation of Transport and 
Shoaling Processes in the Mixing Zone of a 
Stratified Estuary (after Mehta and Hayter, 

1931) 3 

1.2 Interactions of Tidal and Estuarial Sediment 
Transport Processes (after Owen, 1977). ...... 8 

2.1 Repulsive and Attractive Energy as a Function 
of Particle Separation at Three Electrolyte 
Concentrations (after van Olphen, 1963) 17 

2.2 Net Interaction Energy as a Function of Particle 
Separation at High Electrolyte Concentration 

(after van Olphen, 1963) 17 

2.3 Net Interaction Energy as a Function of Particle 
Separation at Intermediate Electrolyte 

Concentration (after van Olphen, 1963) 20 

2.4 Net Interaction Energy as a Function of Particle 
Separation of Low Electrolyte Concentration 

(after van Olphen, 1963) 20 

2.5 Comparison of the Collision Functions for 
Brownian, Shear and Differential Sedimentation 
Coagulation (after Hunt, 1980) 25 

2.6 Variation of SAR with Salinity (Sea Salt 
Concentration) (after Ariathurai, 1974) 27 

2.7 Coagulation-Dispersion Boundary Curves for (a) 
Montmorillonite, (b) II lite and (c) Kaolinite at 

Three pH Ranges (after Kandiah, 1974) 27 

2.8 Monthly Salinity Distributions in the Cumbarjua 
Canal, Goa, India; Ebb; Flood 

(after Rao etal_., 1976) 31 

2.9 Variation in Chloride Concentration in San 
Francisco Bay and Sacramento-San Jaoquin Delta- 
September 1955 (after Orlob et_al_., 1967) 32 

ix 



FIGURE 



PAGE 



2.10 Salinity of the Surface Waters of the Pamlico 
River Estuary as a Function of the Distance from 
the Railroad Bridge in Washington, D.C. (after 

Edzwald et al_., 1974) 33 

2.11 Computed Longitudinal Salinity Profile in the 
Yangtze River Estuary as a Function of the 
Distance Downstream from Jiang Zhen Dong for 
Two River Discharges (after Huang et al . , 

1980) 33 

2.12 A Plot of Raw Viscometer Data Obtained from the 
U.S. Army Corps of Engineers Philadelphia 

District Sample (after Krone, 1963) 39 

2.13 Time and Depth Variation of Suspended Sediment 
Concentration in Savannah River Estuary (after 

Krone, 1972) 45 

2.14 Schematic Representation of the Physical States 
of Cohesive Sediment in Estuary Mixing Zone 

(after Mehta et_al_., 1982a) 46 

3.1 Coordinate System 48 

3.2 Measured Red Density Profiles for Thames Mud 
for Two Different Consolidation Times (after 

Owen, 1970) 57 

3.3 Measured Bed Density Profiles for Avonmouth Mud 
for Two Different Initial Suspended Sediment 
Concentrations (after Owen, 1970) 58 

3.4 Measured Bed Density Profiles for Avonmouth Mud 
for Different Bed Thicknesses (after Owen, 

1970) 59 

3.5 Dimension! ess Density Profiles of Mud Beds 

(after Thorn and Parsons, 1980) 59 

3.5 Variation of Bed Density with Depth for Three 
Different Conditions of Flow-Deposited Beds 
(after Parchure, 1980) 61 

3.7 Dimensionless Density Profile of Mersey Mud with 

Tdc = 2 hours (after Bain, 1981) 62 

3.8 Dimensionless Bed Density Profile of a Mud Bed 

(after Thorn, 1981) . 62 

3.9 Dimensionless Density Profiles for Kaolinite 

Beds with T^c = 2, 5, 11 and 24 hours 63 

X 



FIGURE 



PAGE 



3.10 Dimensionless Density Profiles for Kaolinite 

Beds with T^^, = 48, 72, 96, 144 and 240 hours ... 63 

3.11 Bed Shear Strength Profiles for Kaolinite Beds 

(after Parchure, 1980) 65 

3.12 Bed Shear Strength Profile for a Kaolinite Bed 

(after Dixit, 1982) 66 

3.13 Dimensionless Bed Density Profiles for Salinities 

of 0, 1, 2, 5 and 10 ppt 69 

3.14 Bed Shear Strength Profiles as Functions of 

Salinity 70 

3.15 Bed Schematization Used in Bed • Formation 

Algorithm 72 

3.16 Hypothetical Shear Strength Profile Illustrating 
Determination of Bed Layers Thicknesses 72 

3.17 Laboratory Determined Relationship Between 
Erosion Rate, e, and Bed Shear Stress, t:^ 

(after Mehta, 1981) 80 

3.18 Example of Relationship Between £ and Stress Stress 

(after Mehta, 1981) 80 

3.19 £ - ^^^^ °f Partheniades (1962), Series I 

and II (after Mehta, 1981) 81 

3.20 Dimensionless ^ - \ Relationship Based on 
Results of Ariathurai and Arulanandan (1978) 

(after Mehta, 1981) 83 

3.21 Relative Suspended Sediment Concentration Versus 
Time for a Stratified Bed (after Mehta and 
Partheniades, 1979) 83 

3.22 Schematic Representation of the Selected 
Methodology for the Variation of the Applied 
Bed Shear Stress During Bed Preparation and 
Resuspension Tests (after Mehta et al . , 

1982a) 86 

3.23 Variation of Suspension Concentration with Time 

for Tj^ = 48 Hours (after Dixit, 1982) 87 

3.24 C(T^') Versus t^- for Three Values of Tj^,, 
Using Kaolinite in Salt Water (after Mehta 

et_al_., 1982a) 88 

xi 



FIGURE PAGE 

3.25 Normalized Rate of Erosion, e^-/£ Versus 

Normalized Excess Shear Stress, (zi^) )/ 

T (zu). Using Kaolinite in Tap Water (after 

Mehta et_ al_., 1982a) 90 

3.25 Normalized Rate of Erosion, Versus 

Normalized Excess Shear Stress, ^'^i~'^A^h'^^ ^ 
^(^U^), Using Kaolinite in Salt Water (after 
Mehta et al_. , 1982a) 91 

3.27 Critical Shear Stress Versus SAR for 
Montmorillonitic Soil (after Alizadeh, 

1974) 94 

3.28 Resuspension Rate Versus Normalized Excess Shear 

Stress 97 

3.29 Slope, a , Versus Depth Below Bed Surface, z^, 

as a Function of Salinity 98 

3.30 Ordinate Intercept, e^. Versus Depth Below Bed 

Surface, z^, as a Function of Salinity 98 

3.31 The Internal Circulation Driven by the River 
Discharge in a Partially Stratified Estuary, 
(a) A Transverse Section, (b) A Vertical 

Section (after Fischer et_ al_., 1979) 110 

3.32 Illustration of Wind-induced Circulation (after 
Fischer, 1972) 110 

3.33 Ratio C/Cg Versus Time t for Kaolinite in 
Distilled Water (after Mehta and Partheniades, 

1975) 120 

3.34 Ratio Cg /C^ Versus Bed Shear Stress v (after 

Mehta and Partheniades, 1975) 120 

3.35 Relative Steady State Concentration C^g in 
Percent Against Bed Shear Stress Parameter \-i 

(after Mehta and Partheniades, 1975) 122 

3.36 C in Percent Versus t/t5Q for Kaolinite in 
Distilled Water (after Mehta and Partheniades, 

1975) 124 

3.37 Log t^q Versus for Kaolinite in Distilled 

Water (after Mehta and Partheniades, 1975) 126 

3.38 cj^ Versus for Kaolinite in Distilled 

Water (after Mehta and Partheniades, 1975) 126 

xi i 



FIGURE PAGE 

3.39 Settling Velocity, W^, Versus Suspended Sediment 
Concentration, C, for San Francisco Bay Mud 

(after Krone, 1962) 127 

3.40 Settling Velocity, W^, Versus Suspended Sediment 
Concentration, C, for Yangtze River Estuary Mud 

(after Huang etal_., 1980) 129 

3.41 Versus C for Severn Estuary Mud 

(after Thorn, 1981) 129 

3.42 Effect of Size and Settling Velocity of Elementary 
Particles on the Coagulation Factor of Natural 

Muds (after Bellessort, 1973) 132 

3.43 Effect of Salinity on Settling Velocity of San 
Francisco Bay Mud (after Krone, 1962) 134 

3.44 Effect of Salinity on Settling Velocity of 

Avonmouth Mud (after Owen, 1970) 136 

3.45 Effect of Salinity and Suspension Concentration 
on Settling Velocity of Avonmouth Mud (after 

Owen, 1970) 137 

3.46 Ratio C/Cg Versus Time as a Function of the Bed 
Shear, ti^, for Lake Francis Sediment with 

3=5 ppt 139 

3.47 Ratio C /C^ Versus for Deposition Tests 

with LaRe Francis Sediment 141 

3.48 Apparent Settling Velocity Description in Domains 
Defined by Suspended Sediment Concentration and 

Bed Shear Stress 141 

3.49 Effect of Salinity and Bed Shear Stress on 

Settling Velocity of Lake Francis Sediment 146 

3.50 Settling Velocity Versus Suspension Concentration 

for Deposition Test with Lake Francis Sediment. . . ISO 

3.51 Variation of 0*^ with Salinity and x^ 150 

3.52 Variation of Mean Bed Density with Consolidation 

Time (after Dixit, 1982) 157 

3.53 Variation of p/p^ ^^-^h Consolidation Time (after 

Dixit, 1982) 157 



xi ii 



FIGURE 



PAGE 



3.54 z^/W Versus p/p for Avonmouth, Brisbane, 
Grangemouth and Belewan Muds (after Dixit, 

1982) 158 

3.55 Zu/H Versus p/p for Consolidation Times (a) Less 
TFian 48 Hours and (b) Greater Than 48 Hours 

(after Dixit, 1982) 159 

3.56 Normalized Bed Density Profiles for Thames Mud for 

Two Different Consolidation Times 160 

3.57 Normalized Bed Density Profiles for Avonmouth Mud 

as a Function of Salinity 160 

3.58 Normalized Bed Density Profiles for Avonmouth Mud 

for Different Bed Thicknesses 161 

3.59 Variation of ■^r^zu) with Zj^ for Various 

Consolidation Periods (after Dixit, 1982) 165 

3.60 Correlation of Bed Shear Strength with Bed 

Density (after Owen, 1970) 167 

3.61 Variation of p(Z|^) with Incorporated in 
Consolidation Algorithm 172 

3.62 Bed Schematization Used in Bed Formation - 
Consolidation Algorithms 176 

4.1 Global and Local Coordinates 190 

5.1 Downstream View of Recirculating Flume. Width 
Reducing Device is Shown on Right Side of 

Flume 205 

5.2 Schematic Diagram of Recirculating Flume (after 

Dixit, 1982). 206 

5.3 Kent Mini-Flow Current Meter 208 

5.4 Calibration of Kent Mini-Flow Current Meter .... 208 

5.5 Instrumentation Cart and Setup of Kent Mini-Flow 

Meter and Two Point Gages 210 

5.6 Electric Point Gage and Tube of Water Surface 

Elevation Measuring Device 211 

5.7 Setup of Water Surface Elevation Measuring 

Device (after Wang, 1983) 211 



xi V 



FIGURE PAGE 

5.8 (a) Apparatus I for Obtaining Sediment Core; 

(b) Apparatus II for Sectioning a Frozen Sediment 

Core (after Parchure, 1980) 213 

5.9 Water Sampling Device 215 

5.10 Grain Size Distribution of Kaolinite Used for 

the Experiments 215 

5.11 Shear Stress History for Experiments in the 
Recirculating Flume 219 

5.12 Measurement Stations in the Recirculating 

Flume 221 

5.13 Measured and Predicted Suspended Sediment 
Concentrations for Test No. 1 223 

5.14 Measured and Predicted Suspended Sediment 
Concentrations for Test No. 2 224 

5.15 Measured and Predicted Suspended Sediment 
Concentrations for Test No. 3 225 

5.16 Measured Bed Density Profiles for Experiments in 

the Recirculating Flume 228 

5.17 Rotating Annular Flume 229 

5.18 Velocity-Time Record, and the Measured and 
Predicted Suspended Sediment Concentrations in 

the Tidal Cycle Experiment 232 

5.19 Finite Element Grid of Recirculating Flume; 

Distorted Sketch - Width: Length =4.1:1.0 234 

5.20 Schematic Representation of Set up for Experiment 
in the 100 m Flume (after Dixit et al . , 

1982) 237 

5.21 Measured and Predicted Deposit Thickness Along 

100 m Flume .... 239 

5.22 Aerial View of Camachee Cove Marina 247 

5.23 Bathymetry of Camachee Cove Marina Obtained in 
September, 1982 (after Srivastava, 1983). ..... 248 

5.24 Finite Element Grid of Camachee Cove Marina .... 250 

5.25 Predicted Sedimentation Contours for Marina 

Basin 251 

XV 



t^IGURE PAGE 

5.26 Plan View of 10 km Hypothetical Canal 253 

5.27 Predicted Suspended Sediment Concentration-time 
Record for Element No. 4 in Hypothetical 

Canal 255 

5.28 Predicted Suspended Sediment Concentration-time 
Record for Element No. 5 in Hypothetical 

Canal 256 



xvi 



Abstract of Dissertation Presented to the Graduate Council 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 



PREDICTION OF COHESIVE SEDIMENT MOVEMENT 
IN ESTUARIAL WATERS 

By 

Earl Joseph Hayter 
December 1983 

Chairman' Dr. B.A. Christensen 

Co-Chairman : Dr. A.J. Mehta 

Major Department: Civil Engineering 

Fine sediment related problems in estuaries include shoaling in 
navigable waterways and water pollution. A two-dimensional (horizontal) 
fine, cohesive sediment transport model using the finite element method 
has been developed to predict the temporal and spatial variations of the 
depth-averaged suspended sediment concentrations in estuarial waters. 
The advection-di spersion equation with appropriate source/sink terms is 
solved by the Galerkin weighted residual method for the suspension 
concentration at each node. Contemporary laboratory and field evidence 
has been used to develop algorithms which describe the processes of 
erosion, dispersion, settling, deposition, bed formation and bed 
consolidation. 

XV i 1 



The model yields stable and converging solutions. A useful feature 
of the model is its ability to predict the influence of salinity on the 
rate of fine suspended sediment movement. Verification was carried out 
against results from a series of erosion-deposition experiments in the 
laboratory using kaolinite and a natural mud as the sediments. The 
model was applied under prototype conditions to simulate sedimentation 
in a marina basin and suspended sediment transport in a hypothetical 
canal in which both erosion and deposition occurred. 



XV i i i 



CHAPTER I 
INTRODUCTION 



1.1. Estuarial Cohesive Sediment Dynamics 

Cohesive sediments in estuaries are comprised largely of 
terrigenous clay-sized particles. The remainder may include fine silts, 
biogenic detritus, algae, organic matter, waste materials and sometimes 
small quantities of very fine sand. Although in water with a very low 
salinity (less than about 1 part per thousand) the elementary sediment 
particles are usually found in a dispersed or "non-salt flocculated" 
state, small amounts of salts are sufficient to repress the 
electrochemical surface repulsive forces between the elementary 
particles, with the result that the particles coagulate to form much 
larger aggregates. Each aggregate may contain thousands or even 
millions of elementary particles. The transport properties of 
aggregates are affected by the hydrodynamic conditions and by the 
chemical composition of the suspending fluid. Most estuaries contain 
abundant quantities of cohesive sediments which usually occur in the 
coagulated form in various degrees of aggregation. Therefore, an 
understanding of the transport properties of these sediments in 
estuaries requires a knowledge of the manner in which the aggregates are 
transported in these waters. 

Cohesive sediment transport in estuaries is a complex process 

involving a strong coupling between tides, baroclinic circulation and 

the coagulated sediment. For an extensive description of this process, 

1 



2 

the reader is referred to Postma (1967), Parthem'ades (1971), Barnes and 
Green (1971), Krone (1972), Kirby and Parker (1977), and Kranck 
(1980). In Fig. 1.1, a schematic description is given. The case 
considered is one in which the estuary is stratified, and a stationary 
saline wedge is formed as shown. Various phases of suspended fine 
sediment transport are shown, assuming a ti daily-averaged situation. In 
the case of a partially mixed estuary, the description will be modified, 
but since relatively steep vertical density gradients are usually 
present even in this case, the sediment transport processes will 
generally remain qualitatively similar as depicted in Fig. 1.1. 

As indicated in Fig. 1.1, riverborne sediments from upstream fresh 
water sources arrive in the mixing zone of the estuary. The 
comparatively high degree of turbulence, the associated shearing rates 
and the increasingly saline waters will cause aggregates to form and 
grow in size as a result of frequent interparticle collisions and 
increased cohesion. The large aggregates will settle to the lower 
portion of the water column because of their high settling velocities. 
Results based on laboratory experiments show that aggregate settling 
velocities can be up to four orders of magnitude larger than the 
settling velocities of the elementary particles (Bellessort, 1973). 
Some of the sediment will deposit and some will be carried upstream near 
the bottom until periods close to slack water when the bed shear 
stresses decrease sufficiently to permit deposition. The sediment will 
start to undergo self-weight consolidation. The depth to which the new 
deposit scours when the currents increase after slack will depend on the 
bed shear stresses imposed by the flow and the shear strength of the 
deposit. Net deposition, i.e. shoaling, will occur when the bed shear 



3 




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during flood, as well as during ebb, is insufficient to resuspend all of 
the material deposited during preceding slack periods. Some of the 
sediment that is resuspended will be re-entrained throughout most of the 
length of the mixing zone to levels above the salt water-fresh water 
interface and will be transported downstream to form larger aggregates 
once again, and these will settle as before. At the seaward end some 
material may be transported out of the system, a portion or all of which 
could ultimately return with the net upstream bottom current. 

In the mixing zone of a typical estuary the sediment transport 
rates often are an order of magnitude greater than the rate of inflow of 
"new" sediment derived from upland or oceanic sources. The estuarial 
sedimentary regime is characterized by several periodic (or quasi - 
periodic) macro-time-scales, the most important of which are the tidal 
period (diurnal, semi-diurnal, or mixed) and one-half the lunar month 
(spring-neap-spring cycle). The first is of course the most important 
since it is the fundamental period which characterizes the basic mode of 
sediment transport in an estuary. The second is important from the 
point of view of determining net shoaling rates in many cases of 
engineering interest. 

1.2. Sediment Related Problems in Estuaries 

Estuaries are often centers of population and industry, and as such 
are used as commerce routes to the sea, convenient dump sites for waste 
products as well as for man's recreational enjoyment. They also serve 
as the sink for sediment and pollutants transported by rivers from 
inland sources. As man's activity in and hence dependence upon 
estuaries has increased with the growth of population and commerce in 



5 

these areas, the necessity of proper estuan'al management becomes very 
ostensible. Included in estuarial management is the maintenance of 
navigable waterways and water pollution control, both of which are 
affected to varying degrees by the load of suspended and deposited 
sediment. These two tasks are examined next. 

Under low flow velocities, sometimes coupled with turbulent 
conditions which favor the formation of large aggregates, cohesive 
sediments have a tendency to redeposit in areas such as dredged cuts and 
navigation channels, in basins such as harbors and marinas, and behind 
pilings placed in water (Einstein and Krone, 1962; Ariathurai and F^ehta, 
1983). In addition, as noted previously, the mixing zone between upland 
fresh water and sea water in estuaries is a favorable site for bottom 
sediment accumulation. Inasmuch as estuaries are often used as 
transportation routes, it is desirable to be able to accurately estimate 
the amount of dredging required to maintain navigable depths in these 
water bodies, and also to predict the effect of new estuarial 
development projects such as the construction of a port facility or 
dredging of additional navigation channels. 

Cohesive sediments may influence water quality by affecting aquatic 
life and by providing a large assimilative capacity as well as 
transporting mechanism for dissolved and suspended pollutants. 
Turbidity caused by suspended sediment particles restricts the 
penetration of light, and therefore reduces the depth of the photic 
zone. This in turn may result in a decrease in the production of 
phytoplankton and other algae which leads to a reduction in the amount 
of food available for fish. Deposited sediments can damage spawning 
areas for fish and eliminate invertebrate (e.g. oysters) populations. 



6 

The bulk of the pollution load in a water body is quite often 
transported sorbed to cohesive sediments rather than in the non-sorbed 
state (Preston etal_., 1972; Kirby and Parker, 1973). Therefore, the 
importance of considering the movement of cohesive sediments in 
predicting the fate of pollutants (e.g. pesticides, radioisotopes, and 
toxic elements such as lead, mercury, cadmium, nickel and arsenic) 
introduced in an estuary cannot be overemphasized. The properties of 
cohesive sediments, and in particular clays, which cause the sorption of 
pollutants are the large surface area to volume ratio, the net negative 
electrical charges on their surfaces and their cation exchange 
capacity. These properties are discussed in Chapter II. 

In an investigation of the bottom sediments from several coastal 
marinas in Florida, two interesting observations were made (Weckmann, 
1979; Bauer, 1981). First, when comparing sediment particle size inside 
the basin with that obtained immediately outside in the main body of 
water, it was found that in the majority of the marinas investigated, 
the sediment inside was measurably finer than that outside. Second, a 
similar comparison in terms of heavy metal (e.g. Cu, Pb, Ni, Cd and Zn) 
content within the basin and without indicated measurably higher 
concentrations inside the basin. These two observations, when 
correlated, exemplify the role of cohesive sediments in accumulating 
pollutant levels in estuarial depositional environments such as marina 
basins. This assimilation of pollutants and storage in bottom sediments 
may prove to be an acceptable means of waste disposal, providing the 
contaminated bottom sediment is not resuspended and the pollutant 
desorbed. However, sometimes even a relatively small change in the 
chemical composition of the water may cause desorption of pollutants 
from the sediment particles. 



7 

1.3. Approach to the Problems 

Prediction of the fate of sorbed pollutants or the frequency and 
quantity of dredging required to maintain navigable depths in a channel 
or harbor can be accomplished by modeling the movement of cohesive 
sediments in the water body of concern. It becomes necessary to 
simulate the various transport processes, i.e. erosion, advective and 
dispersive transport, aggregation, deposition and consolidation, and the 
physical factors, e.g. movement of water and dissolved salt, that govern 
these processes. The movement of suspended sediment, water and salt are 
highly interrelated, as is evident upon examination of Fig. 1.2 which 
defines possible interactions between these constituents in an estuary. 

Physical and mathematical models or combinations (hybrid approach) 
of these two types are the types of models available for use in 
predicting cohesive sediment movement in a water body. Physical scale 
models have only been partially successful due to lack of an appropriate 
model sediment as well as due to poor model reproduction of estuarial 
mixing processes and internal shear stresses (Owen, 1977). Mathematical 
models, however, have been generally more successful in reproducing, 
with some degree of accuracy, the movement of cohesive sediments in 
estuarial waters. The modeling philosophy is delineated below. 

To mathematically model the motion of the three main constituents 
in an estuarial environment the three-dimensional forms of the 
conservation of momentum and mass equations for the water and the 
conservation of mass equations for the dissolved salt, suspended 
sediment and pollutant, if present, must be solved numerically. However, 
due to the current high cost of solving such three-dimensional, coupled, 
partial differential equations, only a few three-dimensional models 



8 





Gradual 
Large 



Motion of 
dissolved 
sal t 



Rapid 
Large 



Large 



Bulk flow 
tidal 
propagation 



Rapid 
Large 

Rapid 
aMedi um 



Large 




Rapid 



Velocity field. 
Internal water 
ci rculation. 
Bed shear 



Rapid 
Medium 




Rapid 
Large 



Rapid 



Large 



Medium 



Gradual 
Medi um 



Large 



Rapid 



Very 
Gradual 

Smal 1 




Motion of 
suspended 
mud 



Larae 



Rapid 
Large 



Rapid 
Large 



Rapid 



Medi um 



Coagulation' 
and 

settling /Medium 




Gradual 
Medium 



Fig. 1.2. Interactions of Tidal and Estuarial Sediment Transport 
Processes (after Owen, 1977). 



9 

exist (L1u and Leendertse, 1978). The common procedure has been to 
spatially integrate these equations, laterally and/or vertically, in 
order to reduce them to their two- or one-dimensional forms. 

The horizontal length scales relative to the transport of cohesive 
sediments typically are one to three orders of magnitude greater than 
the vertical length scales in most estuaries. As a consequence, and 
because horizontal transport distances are usually of primary interest 
in ascertaining the magnitude of sedimentation or the fate of sorbed 
pollutants, it is in most cases not unreasonable to use vertically 
integrated transport equations for modeling purposes. However, even 
using the two-dimensional forms of the governing equations, some eight 
to ten coupled equations must be solved to completely model the depth- 
averaged motion of water, sediment and salt. As a result, the modeling 
of water and salt movement is commonly performed separately from the 
sediment transport modeling. For example, a two-dimensional 
hydrodynamic model, which solves the coupled momentum and (water and 
dissolved salt) continuity equations, is used to model the movement of 
water and salt. Then a two-dimensional cohesive sediment transport 
model would be used to predict the motion of sediment using the results 
from the hydrodynamic model. 

1.4. Scope of Investigation 

Mathematical descriptions of the physical processes prevalent in a 
binary fluid-sediment system (such as the flux of sediment to and from 
the bed, and the dispersion, aggregation and settling of suspended 
aggregates in a turbulent flow field) that are incorporated in existing 
suspended cohesive sediment transport models, many of which are 



10 

described in Chapter IV, are limited as they use empirical evidence 
based on limited studies conducted prior to the early 1970's. Since 
that time, a considerable amount of experimental research has been 
conducted, partly at the University of Florida, on the various aspects 
of cohesive sediment transport mechanics. Utilization of contemporary 
laboratory experimental and field evidence to develop new algorithms 
which describe the transport processes of erosion, dispersion, settling, 
deposition and bed consolidation would result in a model whose 
predictive capability is measurably improved over that of existing 
model s. 

The intent of this investigation therefore was twofold: 1) to 
develop a two-dimensional depth-averaged cohesive sediment transport 
model using state-of-the-art information on estuarial cohesive sediment 
processes, and 2) to verify this model with results from 
erosion/deposition experiments performed in a 18 m long recirculating 
flume and in an annular rotating channel, and a deposition experiment 
performed in a 100 m long flume. The sediment process information 
incorporated in the model is based on a detailed analysis and 
interpretation of available laboratory and field data. The results of 
this research is presented in the following format. 

In Chapter II, a description of the composition, structure and 
physicochemical properties of cohesive sediments is given. This is 
followed by discussions on estuarial dynamics and sedimentation 
processes. 

Chapter III begins with a description of the governing equations 
for sediment transport in an estuarial environment. Then, previous 
investigations on erosion, dispersive transport, deposition, and the 



11 

structure and consolidation of cohesive sediment beds are described, 
followed by detailed descriptions of the sediment transport algorithms 
developed in this investigation. 

In Chapter IV, the new cohesive sediment transport model is 
described in detail. Included in this chapter is a review of previous 
models, and descriptions of the finite element solution routine and the 
stability and convergence characteristics of the new model. 

In Chapter V, the model is verified using the results from: 1) 
experiments performed by the investigator in a 18 m long recirculating 
flume at the University of Florida, and 2) a deposition experiment 
performed in a 100 m long flume at the U.S. Army Corps of Engineers 
VJaterways Experiment Station in Vicksburg, Mississippi. The model is 
then used to simulate the sedimentation in a coastal marina. Lastly, 
limitations of the modeling approach, as well as possible model 
applications to water quality and sedimentation management problems are 
di scussed. 

In Chapter VI, conclusions from this study and recommendations for 
future research are presented. 

In Appendix A, the two-dimensional depth-averaged form of the 
advection-dispersion equation is derived. In Appendix B, the element 
coefficient and load matrices are given. In Appendix C, a description 
of the computer program, including a user's manual is given. Lastly, in 
Appendix D, the field data collection and laboratory analysis programs 
required to develop the data base for the model are described. 



CHAPTER II 
BACKGROUND MATERIAL 



2.1. Introductory Note 

Understanding the movement of cohesive sediments 1n an estuarlal 
environment requires knowledge pertaining to the physicochemical 
properties of cohesive sediments, estuarial hydrodynamics and sediment 
transport processes. Each of these topics is briefly discussed below. 

2.2. Description and Properties of Cohesive Sediments 
2.2.1. Composition 

As noted in Chapter I, cohesive sediments consist primarily of 
clay-sized material. Such material consists of clay and non-clay 
mineral components and organic material (Grim, 1968). Clay particles 
are generally less than 2 microns {\m) in size. As a result they are 
termed colloids, and in water possess the properties of plasticity, 
thixotropy and adsorption (van Olphen, 1963). Grim (1968) states that 
"the term clay implies a natural, earthy, fine-grained material", and 
that clays are "composed essentially of silica, alumina, and water, 
frequently with appreciable quantities of iron, alkalies, and alkaline 
earths." The most abundant types of clay minerals are kaolinite, 
montmoril lonite, illite, chlorite, vermiculite, and halloysite. 

Non-clay minerals consist of, among others, quartz, carbonates, 
feldspar, and mica (Grim, 1968). This component of clay material is 

generally larger than 2 i^m in size, though this is not so in all clay 

12 



13 



mateHals. It H currently not possible to quantitatively determine, 
with a high degree of accuracy, the amount of non-clay minerals present 
in a clay mateHal. Grim (1958) points out the consequence of this 
1t«ttatfon by stating that "the absence of accurate quantitative methods 
for determining the non-clay mineral components of clay materials 
frequently makes ft impossible m obtain exact data on the chemical 
composition of the clay minerals themselves in such materials." 

The organic material usually present in clay materials may sxfst as 
discrete particles of matter (e.g. wood), as sorbed organic molecoles on 
the surface of the clay ptrtfdes. or' inserted between clay layers 
(Gn-m. 1968). The percentage by weight of organic matter in a clay 
material may be determined through use of a standard analytical 
procedure such as the Walkley-Rlack test (Allison. 1965). 

Additional possible components of clay materials are water-soluble 
salts, and sorbed exchangeable ions and pollutants. Water-soluble salts 
include chlorides, sulfates, alkaline earths, carbonates of alkalies, 
aluminum and iron. The most common exchangeable cations and anions in 
clay materials are. respectively. Ca^^ Mg2^ n\ K*. mH,"^ and Cr, SO^". 
NO3- and P0,3- (Grim. 1968). The presence of pollutants sorbed to the 
surface of clay particles was discussed in Chapter 1. 

2.2. g. Origin 

In nature clay materials are produced by hydrothermal action and 
weathering of rocks. Fact0rs which Influence the type of clay minerals 
formed by these two processes include the composition of the parent 
rock, m Climate, topography, the abundance and kind of vegetition, 
time, and pH of the ground water, the presence and kind of alkalies and 



14 

alkaline earths, the intensity of the hydrothermal alteration, and the 
permeability and porosity of the host rock. Grim (1968) describes the 
origin and occurrence of clay materials in detail. 

2.2.3. Structure 

Clay minerals are primarily hydrous aluminum silicates with 
magnesium or iron occupying all or part of the aluminum positions in 
some clays, and with alkalies (e.g. sodium, potassium) or alkaline 
earths (e.g. calcium, magnesium) also present in others (Grim, 1968). 
Most clays are composed of one of two atomic structural units, or 
combinations of the two basic units. These are the silica tetrahedron 
and the aluminum hydroxide octahedral unit. The former consists of a 
central silicon atom surrounded by four oxygen atoms or hydroxy Is 
arranged in a tetrahedral configuration. The tetrahedrons are bonded 
together in a hexagonal network in such a way that a sheet structure of 
composition Si40g(0H)4 is formed. The structure of each sheet is such 
that the oxygens and/or hydroxy Is forming the bases of the tetrahedral 
units are in the same plane, and the tips of all the units point in the 
same direction (Grim, 1968). 

The octahedral aluminum hydroxide unit consists of "two sheets of 
closely packed oxygens or hydroxyls in which aluminum, iron, or 
magnesium atoms are embedded in octahedral coordination, so that they 
are equidistant from six oxygens or hydroxyls" (Grim, 1968). If 
aluminum atoms are present, only two out of every three central 
positions will be filled so that the structure's electrical charge will 
be balanced. However, if magnesium atoms are present, all central 
positions are occupied. With aluminum, the octahedral is known as the 



15 



gibbsite structure, which has the formula Alj(OH)g. The brucite 
structure is that formed with magnesium, and has the formula Mg3(0H)g 
(Grim, 1968). 

The different clay minerals are formed by stacking of the sheet 
structures to form layers, and substitution of different ions for the 
aluminum in the octahedral unit. Grim (1968) gives an excellent 
description of the structural arrangement of the most common types of 
clay minerals. 

Ions of one kind are sometimes substituted by ions of another kind, 
with the same or different valence. This process does not necessarily 
involve replacement. The tetrahedral and octahedral cation 
distributions develop during initial formation of the mineral, and not 
by later substitution (Mitchell, 1976). Substitution in all the clay 
materials, except for kaolinite, gives clay particles a negative 
electric charge which is of great significance in coagulation of clays 
and in absorption of pollutants. Another cause of net particle charge 
is the preferential sorption of peptizing ions on the surface of the 
particle (van Olphen, 1963). 

2.2.4. Interparticle Forces 

For particles in the colloidal size range, surface physicochemical 

forces exert a distinct influence on the behavior of the particles due 

to the large specific area, i.e. ratio of surface area to volume. As 

stated previously, most clay particles fall within the colloidal range 

in terms of both their size (2 lim or less) and the controlling influence 

of surface forces on the behavior. In fact, the average surface force 

6 

on one clay particle is approximately 10 times greater than the 
gravitational force (Partheniades, 1962), 



16 

The relationship between clay particles and water molecules is 
governed by the interparticle electrochemical forces. The different 
configurations and groupings as well as electric charges of clay 
particles affect their association with water molecules (Grimshaw, 
1971). Water molecules possess a permanent electrical imbalance or 
dipole moment which results from the molecular arrangement of the oxygen 
and hydrogen atoms. According to Grim (1962), the electrostatic field 
emanating from the surface of a clay particle orients the polar water 
molecules in the pores separating adjacent particles. 

Interparticle forces consist of both attractive and repulsive 
forces. The attractive forces present are the London-van der Waals, and 
are due to the nearly instantaneous fluctuation of the dipoles which 
result from the electrostatic attraction of the nucleus of one atom for 
the electron cloud of a neighboring atom (Grimshaw, 1971). These 
electrical attractive forces are weak, and are only significant when 
interacting atoms are very close together. However, they are strong 
enough to cause structural build-up as they are additive between pairs 
of atoms. Thus, the total attractive force between two clay particles 
is equal to the sum of the attraction between all the atoms comprising 
both particles. This additive effect results in a larger attractive 
force and to a smaller decrease in this force with increasing particle 
separation. Figure 2.1 shows qualitatively the relationship between the 
attractive energy V^^ of one particle for another and the particle 
separation distance. The attractive energy is inversely proportional to 
the sixth power of the separation distance for two atoms and to the 
second power for two spherical particles. The magnitude of decreases 
with increasing temperature and is dependent upon the geometry and the 



17 




Double -Layer Repulsion at 
Three Different Electrolyte 
Concentrations 



\ \ Internnediate 
VliqtNN^ Particle Seoar a t ion 



van der wools 
Attraction 




Fig. 2.1. Repulsive and Attractive Energy as a Function of Particle 
Separation at Three Electrolyte Concentrations (after 
van Olphen, 1953) . 



V 



Net Interaction 

Energy 



Particle Seoaration 




mm 



Fig. 2.2. Net Interaction Energy as a Function of Particle Separation 
at High Electrolyte Concentration (after van Olphen, 1953). 



18 

size of adjacent clay particles. The attractive energy has been found 
to be only slightly dependent upon the salt concentration (i.e. 
salinity) of the medium (van Olphen, 1963). 

The repulsive forces of clay materials are due to the negatively 
charged particle forces. The repulsion potential increases in an 
exponential fashion with decreasing particle separation. The magnitude 
of these forces is dependent upon the salinity, decreasing with 
increasing salinity as shown in Fig. 2.1, where is the repulsive 
energy. This dependence of Vj, on the salinity can best be explained 
using the concept of the electrical double layer and the surrounding 
diffuse layer, van Olphen (1963) states that the double layer is 
composed of the net electrical charge of the elementary clay particle 
and an equal quantity of ionic charge of opposite sign located in the 
medium near the particle surface. Thus, the net electrical charge is 
balanced in the surrounding medium. The ions of opposite charge are 
called the counter-ions, i.e. cations. The counter-ion concentration 
increases with decreasing distance from the particle surface. This 
layer of counter-ions is referred to as the diffuse layer. A clay 
particle and the associated double layer is referred to as a clay 
micelle (Partheniades, 1971). When the salinity is increased, the 
diffuse layer is compressed toward the particle surface (van Olphen, 
1963). The higher the salinity, and as well the higher the valence of 
the cations which compose the diffuse layer, the more this layer is 
compressed and the greater the repulsive force is decreased. 

With a high salinity, corresponding to a value approximately that 
of seawater (35 ppt), the attractive forces become predominant at all 
but extremely small particle separation distances. The interaction 



19 

potential, determined by summing and V^, reflects this dominance, and 
shows the highest attractive potential (primary minimum) at separation 
distances on the order of 1 nm (10~\) (Parker, 1980). At distances 
less than this the short range repulsive forces are predominant (van 
Olphen, 1963). Figure 2.2 shows this net interaction potential as a 
function of particle separation for high salinity. Thus, two clay 
particles will adhere when they reach the separation distance at which 
the primary minimum occurs. Cohesion or particle destabi 1 ization occurs 
at a maximum rate due to the presence of attractive forces even at 
relatively great distances. 

For medium and low salinities, on the order of 10-15 ppt and 1-2 
ppt respectively (Parker, 1980), repulsive forces become predominant at 
separation distances of approximately 10 nm where a local repulsive 
potential maximum occurs (Figs. 2.3 and 2.4). At distances closer than 
this, these interaction potentials are similar to that for high 
salinity. As indicated by these figures, the destabi 1 ization of two or 
more particles would be expected to decrease for decreasing salinities 
as a result of net repulsive forces existing at increasingly larger 
distances (van Olphen, 1963). 

2.2.5. Cation Exchange Capacity 

The cation exchange capacity (CEC) is an important property of 
clays by which they sorb certain cations and anions in exchange for 
those already present and retain them in an exchangeable state. The CEC 
of different clays varies from 3-15 milliequivalents per 100 grams of 
material (meg/100 gm) for kaolinite to 100-150 meg/100 gm for 
vermiculite. Higher CEC values indicate greater capacity to absorb 



20 




Fig. 2.3. Net Interaction Energy as a Function of Particle Separation 
at Intermediate Electrolyte Concentration (after van Olphen, 
1953). 




T min 

Fig. 2.4. Net Interaction Energy as a Function of Particle Separation 
at Low Electrolyte Concentration (after van Olphen, 1963). 



21 

other cations. The negative surface charge caused by isomorphous 
substitution is neutralized by sorbed cations located on the surfaces 
and edges of a clay particle. These cations remain in an exchangeable 
position and may in turn be replaced by other cations. 

The following factors are the causes of cation exchange: 1) 
substitution within the lattice structure results in unbalanced 
electrical charges in the structural units of some clays, and 2) broken 
bonds around the edges of the tetrahedral -octahedral units give rise to 
unsatisfied charges. In both cases the unbalanced charges are balanced 
by the sorbed cations. The number of broken bonds and hence the CEC 
increases with decreasing particle size. 

The ability to replace exchangeable cations depends on the 
concentration of the replacing cation, the number of available exchange 
positions, the nature of the anions and cations in the replacing 
solution. Increased concentration of the replacing cation results in 
greater cation exchange. The release of an ion depends upon the nature 
of the ion itself, upon the nature of the other ions filling the 
remaining exchange positions, and upon the number of unfilled exchange 
sites. The higher the valence of a cation, the greater is its replacing 
power and the more difficult it is to displace when sorbed on a clay. 
Some of the predominantly occurring cations in sediments are sodium, 
potassium, calcium, aluminum, lead, copper, mercury, chromium, cadmium 
and zinc. 

2.2.6. Coagulation 

Coagulation of suspended cohesive sediments depends upon 
interparticle collision and interparticle cohesion after collision. 



22 

Cohesion and collision, discussed in detail by among others Kruyt 
(1952), Einstein and Krone (1962), Krone (1962), Partheniades (1964), 
O'Melia (1972), and Hunt (1980) are reviewed here. 

The collision frequency, I, for suspended sediment particles of 
effective diameters d^- and d - is given by (Hunt, 1980): 



I = P(d.,d ) dN dN. (2.1) 

'J 'J 



where P(d^-,dj) = collision function determined by the collision 
mechanism (discussed below), which has units of fluid volume per unit 
time, and dN^ = number of particles with sizes between d^- and d^-+d(d^-) 
per unit volume of the fluid. 

There are three principle mechanisms of interparticle collision in 
suspension, and these influence the rate at which elementary sediment 
particles coagulate. The first is due to Brownian motion resulting from 
thermal motions of molecules of the suspending ambient medium. The 
collision function corresponding to this mechanism is given by (Hunt, 
1980): 

2 kT. (d.+d.)^ 

Pu(d.,d.) = ^ ' ^ (2.2) 

° ^ 3 ^ d.d. 

where k = Boltzmann constant, T|^ = absolute temperature and \^ = dynamic 
viscosity of the fluid. Generally, coagulation rates by this mechanism 
are too slow to be significant in estuaries unless the suspended 
sediment concentration exceeds 10 g/1 . Aggregates formed by this 
mechanism are weak, with a lace-like structure and are easily fractured 
by shearing in the flow or are crushed easily when deposited (Krone, 
1962). 



23 



The second mechanism is that due to internal shearing produced by 
local velocity gradients in the fluid. Collision will occur if the 
paths of the particle centers in the velocity gradient are displaced by 
a distance which is less than the sum of their radii (referred to as the 



where G is the local shearing rate and R^- j = d^.+dj. Aggregates produced 
by this mechanism tend to be spherical, and are relatively dense and 
strong because only those bonds that are strong enough to resist the 
internal shearing due to local velocity gradients can survive. The 
frequency of collision is especially high in an estuarial mixing region 
where a large number of suspended particles are found. 

The third mechanism, differential sedimentation, results from the 
fact that particles of different sizes have different settling 
velocities. Thus a larger particle, due to its higher settling 
velocity, will collide with smaller, more slowly settling particles 
along its path and will have a tendency to "pick up" these particles on 
its way down. The collision function is expressed as 



collision radius, R^-j, between d^- and dj size particles). The collision 
function is given as 



P3,{d..dj) 



^ 3 
= - R. . 

6 



(2.3) 



Tig p^-p^ 



w 



.)(d.+d.)^|d.^-d.^! 



(2.4) 




11 1 J 



24 

where v = kinematic viscosity of the fluid, = floe density and = 
fluid density. This mechanism produces relatively weak aggregates and 
contributes to the often observed rapid clarification of estuarial 
waters at slack. 

All three mechanisms operate in an estuary, with internal shearing 
and differential sedimentation generally being predominant in the water 
column, excluding perhaps the high density near-bed layer, where 
Brownian motion is likely to contribute significantly as a collision 
mechanism. Then again, internal shearing is probably more important 
than differential sedimentation during times, excluding those near slack 
water, when collision and coherence due to differential settling would 
be expected to be the main mechanism controlling the rate of 
coagulation. 

Hunt (1980) compared the values of the three collision functions 
(Eqs. 2.2 - 2.4) for collision of a d^.=l m size particle with varying 
sizes, dj, of the colliding particle under the following conditions: 
temperature 14°C, shearing rate G = 3 sec"-"-, and (P^'P^^fP^ ~ 0.02. The 
comparison is shown in Fig. 2.5 and reveals that each collision 
mechanism is dominant over a certain particle size range. In this 
example, Brownian motion is the dominant mechanism for particles less 
than 1 m, internal shearing is dominant for particles between 1 and 100 
m, and differential sedimentation is dominant for particles greater 
than 100 \m. Hunt states that the same ordering of the dominant 
collision mechanisms with increasing dj would be achieved for collisions 
with other d^- sizes. Thus, the collision frequency is controlled not 
only by the prevailing flow conditions and local suspension 
concentration, but by the size of the colliding particles and/or floes 
as well. 



25 




d; (/xm) 



Fig. 2.5. Comparison of the Collision Functions for Brownian, Shear 

and Differential -Sedimentation Coagulation (after Hunt, 1980). 



26 



Cohesion or particle destabilization of colloidal particles is 
caused by the presence of net attractive electrochemical surface forces 
on the particles. The latter condition is promoted by increased 
concentration of dissolved ions and/or increased ratio of multivalent to 
monovalent ions, both of which serve to depress the double layer around 
micelles and thus allow the attractive London-van der Waals and 
coulombic forces to predominate (Krone, 1963). Since sea salt is a 
mixture of salts, with monovalent sodium ions and divalent calcium and 
magnesium ions prevalent in natural electrolytes, the effect of these 
salts on cohesion is determined by the relative abundances of mainly 
these three ions (see Table 3.2), the latter being indicated by the 
sodium adsorption ratio (SAR). The SAR is defined as 



where the cation concentrations are in milliequivalents per liter 
(Arulanandan, 1975). The relationship between the SAR and salinity is 
seen in Fig. 2.6. The cation exchange capacity (CEC), salinity and SAR 
all serve to determine the net interparticle force and thus the 
potential for micelles to become cohesive. 

Kandiah (1974) found that the boundary between the dispersed and 
coagulated states for the three main clay groups, kaolinite, illite and 
montmorillonite, varied with the SAR, total salt concentration and pH of 
the solution (see Figs. 2.7a, b, and c). The dashed lines in these 
figures represent interpolated boundary curves for a pH range of 7.5 to 



27 






CO 


CO 






0) 




> 


CO 




































a. 


cr 




ee 


O ^ 
^3 


pu 


rs 






o 








+-> 


NC 


Ull 


+-> • 
ro -— ~ 




•r- 


■=3- 


8 


(/) 




S- 


+-> CTi 




pe 


•1— I— ( 


O 


1/1 


o ^ 


og 






Q 


r— "to 




C 




O 


o c 






_J 


+-> 






rd 


O !- 

s: ID 


o 


13 
C7> 


1- 


oa 


S. M- 

o ro 









ro 



CM 




ro 




<D 
















+-) 








c: 




*i— 


i- 




0) 


ro 


+J 


00 






ro 


-C 




4-> 




•r- 




3 


c • 




o •— - 




•1- 


< 


+-> 




ro a\ 






M- 


-p 


o 






0) -r- 


c 


u ro 


o 


c: s_ 




O 13 


+J 




ro 






+j ro 


tl 




ro 


'ro ^ 


>- oo <: 


UD 









28 






Q. 












CD 




-a 


0) 


— ^ 




S- 






.c 




o 


+J 




m 


+J 


r— 1 








o 










n3 




+J 


•r— 


s_ 




"O 


0) 




c 


Q. 




03 


to 








o 




Q 


03 


i- 






OJ 


C 




+-> 


O 


S- 


q- 


•r— 


o 


03 


-!-> 4- 




n3 






r— 




i/i 






0) 


cn 


> 


CD 


re 


S- 


(— 


o 


r3 


03 




o 





o 




CT 

o 
o 

z 
o 

o < 
_1 

o 



o 



o 



ca s- 1—1 



c +-> 
o 

■1- +-> 

(/) 03 

i- 

O) QJ 

I/) T- 



o3 



Q r— S- 

I O) 

E -P 
O S- 

•I- O 03 

+J <4- V — 
03 

1 — c/i (/) 

13 O) OJ 

cn > cn 

ro S- C 

O 3 03 

O O 



29 

8.5, which 1s the range found in sea water at all salinities. It is 
evident that the boundary between the dispersd and coagulated states for 
these three clays are different. Kaolinite becomes cohesive at a 
salinity of 0.6 pnt, illite at 1.1 ppt and montmorillonite at 2.4 ppt 
(Ariathurai, 1974). Whitehouse et_ al_. (1960) and Edzwald et al_. (1974) 
reported that the cohesiveness of these micelles develops quickly at the 
given salt concentrations, and that little increase in coagulation 
occurs at higher salt concentrations, which implies that the micelles 
must have attained the maximum degree of cohesion. The rapid 
development of cohesion and the low salinities at which the main clay 
types become cohesive indicates that cohesion is primarily affected by 
salinity variations near the landward end of an estuary where salinities are 
less than about 3 ppt. 

The above cohesion mechanism is referred to as salt flocculation. 
There is another cohesion mechanism that operates in water between 
micelles in the absence of salt, and hence it is termed non-salt 
flocculation. However, in an estuary the conditions are conducive for 
destabilization to be caused by salt flocculation (i.e. depression of 
the diffuse double layer). Both types of destabilizing mechanisms are 
reported in detail by Lambe (1953). 

In summary, it is apparent that cohesive sediment transport in 
estuaries is strongly influenced by the coagulation behavior of 
dispersed sediment particles, which is controlled by the salinity field, 
velocity gradients and the concentration of suspended sediments. In 
particular, the salinity of the suspending fluid affects the process of 
coagulation in two ways: 1) elementary clay particles become cohesive 
when the salinity is equal to or greater than 1-3 ppt, and 2) the 



30 



presence of high velocity gradients in the estuarial mixing zone 
increases the collision frequency betv;een dispersed particles and/or 
aggregates. 

2.3. Significance of Important Physical Factors in 

Estuarial Transport 
2.3.1. Estuarial Dynamics 

The hydrodynamic regime in an estuary is governed by the 
interaction between fresh water flow, astronomical tides, wind-generated 
surface waves, surface (i.e. wind) stresses, Coriolis force, the 
geometry of the water body and the roughness characteristics of the 
sedimentary material composing the bed (Dyer, 1973). Geometry includes 
the shape and the bathymetry of the estuary. The geometry and bed 
roughness interact with the driving forces - the first five factors - to 
control the pattern of water motion (in particular the shear stress and 
turbulence structure near the bed), frictional resistance, tidal damping 
and the degree of tidal reflections (Ippen, 1966). 

The magnitude of the tidal flow relative to the fresh water inflow 
governs, to a large extent, the intensity of vertical mixing of the 
lower high density layer with the upper less dense layer. There exists 
in all estuaries a horizontal, i.e. longitudinal, salinity profile which 
decreases from the mouth to the upper reaches of the estuary. Such 
profiles have been measured in numerous estuaries world-wide. A few 
examples included here are: Cumbarjua canal, Goa, India during the dry 
season, i.e. October through June (Fig. 2.8); the San Francisco Bay and 
Sacramento-San Joaquin Delta (Fig. 2.9); the Pamlico River Estuary, 
North Carolina (Fig. 2.10); and the Yangtze River Estuary, China 



31 




Fig. 2.8. Monthly Salinity Distributions in the Cumbarjua Canal, 

Goa, India; Ebb; --- Flood (after Rao et al. , 1975). 



32 



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33 




DISTANCE DOWNSTREAM (km) 



Fig. 2.10. Salinity of the Surface Waters of the Pamlico River Estuary 
as a Function of the Distance from the Railroad Bridge at 
Washington, D.C. (after Edwald et al_. , 1974). 



40 




DISTANCE SEAWARD FROM JIANG ZHEN DOJG(km) 

Fig. 2.11. Computed Longitudinal Salinity Profile in the Yangtze River 
Estuary as a Function of the Distance Downstream from Jiang 
Zhen Dong for Two River Discharges (after Huang et , 1980). 



34 

(Fig. 2. 11). The existence of a longitudinal salinity gradient, or 
baroclinic force, implies that there could be a gravity driven upstream 
transport of a high density sediment suspension in the lower portion of 
the water column (Officer, 1981; Mehta and Hayter, 1981). 

Winds affect the hydrodynamic regime and mixing in an estuary by 
generating a surface shear stress and waves. The surface stress is 
capable of generating a surface current (whose magnitude will be 
approximately three percent of the wind speed at 9.1 m elevation 
(Hughes, 1956)) and a superelevation of the water surface along a land 
boundary located at the downwind end of the estuary (Ippen, 1966). The 
latter effect causes a vertical circulation cell, with landward flow at 
the surface and a reversed seaward flow along the bottom. This 
phenomenon as well increases the degree of vertical mixing. 

Along the banks and in shallow areas, surface gravity waves induced 
by the wind are capable of eroding bottom sediments. Since a tidal 
current of sufficient strength to transport (but not necessarily to 
erode the sediment by itself) suspended sediment is generally present, 
this material is advected and dispersed both longitudinally with the 
main tidal flow and sometimes laterally with secondary currents towards 
the deeper sections of the estuary. Wave action and in particular wave 
breaking substantially increase the intensity of surficial turbulence 
and mixing. 

The Coriolis force, caused by the earth's rotation, has both a 
radial (horizontal) and a tangential (vertical) component. The latter 
is generally negligible as it is linearly proportional to the vertical 
component of the flow velocity, which is typically an order of magnitude 
smaller than the horizontal velocity components. The magnitude of the 



35 



radial component depends upon the size of the water body. Most extra- 
tropical estuaries are relatively large and therefore the effect of this 
force on the hydrodynamic regime is measurable. Estuarial hydrodynamics 
are described in extensive detail in such texts as Ippen (1966), Barnes 
and Green (1971), Dyer (1973), Officer (1976) and Fischer et al . (1979). 

2.3.2. Sediment Processes 

The sedimentary regime in an estuary is controlled by the 
hydrodynamics, the chemical composition of the fluid and the 
physicochemical properties of the cohesive sediment. These factors 
affect the processes of erosion, advection, dispersion, aggregation, 
settling., deposition and consolidation of the deposited bed. These 
processes are briefly described below, following a definition of a clay 
suspension. 

A "solution" of clay in a medium consists of a homogeneous 
dispersion of very small kinetic units, i.e. particles (van Olphen, 
1963). When the Stokes diameter of the clay solution is less than 2 \m, 
the clay dispersion is usually referred to as a sol. The Stokes 
diameter of an arbitrarily shaped particle is determined by equating the 
particle's settling velocity with Stokes law for spherical particles and 
solving for the "equivalent spherical diameter" (Stokes diameter). When 
this diameter is greater than 2 ^m, the dispersion is called a 
suspension. However, through use, the term suspension has become 
synonymous with dispersion, and thus a clay suspension refers to both 
sol and suspension. 

Erosion of cohesive soils occurs whenever the shear stress induced 
by fluid flow over the bed is great enough to break the electrochemical 



36 

interparticle bonds (Partheniades, 1965; Paaswell, 1973). When this 
happens, erosion takes place by the removal of individual sediment 
particles and/or aggregates. This type of erosion is time dependent and 
is defined as surface erosion or resuspension. In contrast, another 
type of erosion occurs more or less instantaneously by the removal or 
entrainment of relatively large pieces of soil. This process is 
referred to as mass erosion or redispersion and occurs when the flow- 
induced shear stresses on the bed exceed the soil bulk strength along 
some deep-seated plane. 

Once eroded from the bed, cohesive sediment is transported entirely 
as suspended load (not as bed load) by the estuarial flow. Such 
transport is the result of three processes: 1) advection - the sediment 
is assumed to be transported at the speed of the local mean flow, 
2) turbulent diffusion - driven by spatial suspended sediment 
concentration gradients, the material is diffused laterally across the 
width of the flow channel, vertically over the depth of flow and 
longitudinally in the direction of the transport, and 3) longitudinal 
dispersion - the suspended sediment is as well dispersed in the flow 
direction by spatial velocity gradients (Ippen, 1966). 

In fresh water, most clay particles are in a stabilized or 
dispersed state because the repulsive electrochemical surface forces 
between the particles prevent them from adhering to one another upon 
collision. In the increasingly saline conditions encountered moving 
seaward in estuaries, the repulsive forces are suppressed and clay 
particles coagulate to form floes. A systematic "build up" of floes as 
occurs in estuaries is defined as aggregation. An aggregate is 
considered to be the structural unit formed by the joining of floes. 



37 



The rate and degree of aggregation are two important factors which 
govern the transport of cohesive sediments in estuaries. Factors, 
besides the water chemistry and the magnitude of the surface forces, 
known to govern coagulation and aggregation include sediment size 
grading, mineralogical composition, particle density, organic content 
and the suspended concentration (i.e. availability) of the sedimentary 
material, the water temperature, height through which the floes have 
settled, and the turbulence intensity (represented by the shearing rate 
G) of the suspending flow (Owen, 1971). 

Given the mechanisms which influence the rate of aggregation in an 
estuary, the order of aggregation, which characterizes the packing 
arrangement, density and shear strength of aggregates, is determined by: 
1) sediment type, 2) fluid composition, 3) local shear field, and 4) 
concentration of particles or floes available for aggregation. With 
regard to the second factor. Krone (1962; 1978) found that the structure 
of aggregates is dependent on the salinity for salinities less than 
about 10 ppt. 

Primary or 0-order floes are highly packed arrangements of 
elementary particles, with each floe consisting of perhaps as many as a 
million particles. Typical values of the void ratio (volume of pore 
water divided by volume of solids) have been estimated to be on the 
order of 1.2. This is equivalent to a porosity of 0.55, which is a more 
"open" structure than commonly occurs in eohesionless sediments (Krone, 
1963). Continued aggregation under favorable shear gradients can result 
in the formation of first or higher order aggregates composed of loosely 
packed arrays of 0-order floes. Each succeeding order consists of 
aggregates of lower density and lower shear strength. Experimental 



38 

observations (Krone, 1963; 1978) tend to indicate the following 
approximate relationship between the aggregate shear strength, t^, and 
aggregate density, p^^, for many (although not all) sediments 

% = (P,-l)^ (2.6) 

where a and p are coefficients which must be determined experimentally 
for each sediment. Inasmuch as the shear field in an estuary exhibits 
significant spatial and temporal variations, a range of aggregates of 
different shear strengths and densities are formed, with the highest 
order determined by the prevailing shearing rate, G = du/dz, provided 
that: 1) the sediment and the fluid composition remain invariant, and 
2) sufficient number of suspended particles are available for promoting 
coagulation and aggregation. 

The determination of -c^ and corresponding to each sediment-fluid 
mixture can be carried out through rheological diagrams of applied shear 
stress against the shearing rate. Such plots were developed by Krone 
(1963; 1978) with the help of a specially designed annular viscometer. 
An example of such a diagram is presented in Fig. 2.12, with the shear 
stress proportional to the dial reading on the viscometer and the 
shearing rate proportional to the rotation rate of the outer cylinder of 
the viscometer. Each order of aggregation corresponds to a given volume 
fraction of the aggregates (volume occupied by the aggregates divided by 
the total volume of the suspension) which in turn can be shown to be 
related to the relative differential viscosity (the viscosity of the 
suspension divided by the viscosity of the suspended medium). Given the 
viscosity of the suspending medium, the relative differential viscosity 



39 



Q. 




a '9Niav3H nvia 



40 

is determined from the slope of the rheological diagram, and hence the 
volume fraction can be calculated. The density is then computed from 
the volume fraction. The intercept on the applied shear stress axis of 
the diagram corresponds to i;^; in Fig. 2.12 the ordinate intercept is 
proportional to t^^. Table 2.1 gives the orders of aggregation, cation 
exchange capacity (CEC), densities and shear strengths of sediment 
samples from five different sources. As observed in this table, the 
first four sediment samples are characterized by three orders of 
aggregation while the sediment from San Francisco Bay is characterized 
by six orders. The number of aggregation orders possible for a 
suspension of a given sediment is equal to the number of linear segments 
on the rheological diagram with different slopes. Thus, in Fig. 2.12, 
the sediment sample has two possible orders of aggregation. Krone 
(1963) postulated that each segment is related to a particular volume 
fraction and therefore to a different manner in which the same sediment 
can aggregate, i.e. different order of aggregation. Thus, for the 
suspensions of the first four sediments listed in Table 2.1, three 
different linear segments were obtained on the rheological diagrams, 
while for Bay mud, six segments, and therefore six orders of aggregation 
were found. This indicates that Bay mud can aggregate in three more 
ways than the other four sediments, and further suggests that the Bay 
sediment is more cohesive than the others. Also observed in this table 
is the very rapid decrease in the shear strengths and somewhat less 
rapid decrease in densities with increasing order of aggregation. These 
trends indicate that as the order of aggregation increases, the inter- 
aggregate pore volume increases and the strength of these aggregates 
decreases because of limited bonding area between the lower order 
aggregates (Krone, 1978). 



41 

Table 2.1 

Properties of Sediment Aggregates (after Krone, 1963) 



Sediment Order of CEC Density Shear Strength 

Sample Aggregation (meq/100 gm) "^"^^ ^s^'^ ^'^^ 



Brunswick 

Harbor 0 38 1164 3.40 

1 1090 0.41 

2 1067 0.12 

3 1056 0.062 

Wilmington 

District 0 32 1250 2.10 

1 1132 0.94 

2 1093 0.25 

3 1074 0.12 

Gulf port 

Channel 0 49 1205 4.60 

1 1106 0.69 

2 1078 0.47 

3 1065 0.18 

VJhite River 

(salt) 0 60 1212 4.90 

1 1109 0.68 

2 1079 0.47 

3 1065 0.19 

San Francisco 

Bay 0 34 1269 2.20 

1 1179 0.39 

2 1137 0.14 

3 1113 0.14 

4 1098 0.082 

5 1087 0.036 

6 1079 0.020 



42 

The settling rate of coagulated sediment particles depends on, in 
part, the size and density of the aggregates and as such is a function 
of the processes of coagulation and aggregation (Owen, 1970). Therefore 
the factors which govern these two processes also affect the settling 
rate of the resulting aggregates. As noted in Chapter I, the settling 
velocities of aggregates can be several orders of magnitude larger than 
those of individual clay particles (Bellessort, 1973). 

Deposition of aggregates occurs relatively quickly during slack 
water. Deposition also occurs in slowly moving and/or decelerating 
flows, as was observed (see Fig. 2.13), for example, in the Savannah 
River Estuary during the second half of flood and ebb flows (Krone, 
1972). Under such conditions only those aggregates with shear strengths 
of sufficient magnitude to withstand the highly disruptive shear 
stresses in the near bed region will actually deposit and adhere to the 
bed. Thus, deposition is governed by the bed shear stresses, turbulence 
structure above the bed, type of sediment, depth of flow, suspension 
concentration and the ionic constitution of the suspending fluid (Mehta 
and Partheniades, 1973). An important conclusion derived from extensive 
laboratory erosion and deposition experiments using a wide range of 
cohesive sediments under steady flow conditions was that under these 
conditions the two processes do not occur simultaneously as they do in 
cohesionless sediment transport (r-lehta and Partheniades, 1975; 1979; 
Parchure, 1983). 

A flow-deposited bed of cohesive sediment aggregates possesses a 
vertical bulk density and shear (i.e. yield) strength profile which 
changes in time primarily due to consolidation. Secondary causes are 
thixotropy and associated physicochemical changes affecting 



43 

interparticle forces. Consolidation, caused by the gravitational force 
(overburden) of overlying deposited aggregates which crushes and thereby 
decreases the order of aggregation of underlying sediment, has been 
observed to occur in three phases (Migniot, 1968). During the first 
phase the bed consolidates quickly as the water in the bed moves upward 
through the interstices of the bed material. This phase has been found 
to last up to approximately 10 hours for cohesive sediments (Owen, 
1977). During the second phase, which can last up to about 500 hours, 
water is expelled from the bed by percolation. The rate of 
consolidation during the third phase is even slower and the length of 
time it takes for a cohesive sediment bed to reach its final, fully 
consolidated state depends upon the nature of the sedimentary material 
comprising the bed and the chemical composition (i.e. ionic 
concentrations) of the bed pore water (Owen, 1977). The average values 
of the bed bulk density and shear strength increase and their vertical 
profiles change during each of these three phases. Consideration of the 
consolidation process is essential in modeling the erosive behavior of 
such beds because: 1) the susceptibility to erosion of a consolidating 
bed decreases with time due to the continual increase in shear strength, 
and 2) the vertical profile of the shear strength determines the level 
to which a bed will erode when subjected to excess shear, i.e. an 
applied bed shear stress in excess of the shear strength of the bed 
surface. 

From an Eulerian point of view, the superposition of oscillating 
tidal flows on the quasi-steady state transport phenomenon depicted in 
Fig. 1.1 results in corresponding oscillations of the suspended sediment 
concentration with time as shown by the Savannah River data in Fig. 



44 

2.13. Such a variation of the suspended load uUimately results from a 
combination of advective and dispersive transport, erosion and 
deposition. 

Because of the complexity of the phenomena, more than one 
interpretation is possible as far as any schematic representation of 
these phenomena is concerned. One such representation is shown in Fig. 

2.14. According to this description, cohesive sediments can exist in 
four different physical states in a tidal estuary or sea: as a mobile 
suspension, a stationary suspension, a partially consolidated bed and as 
a settled bed. The last two are formed as a result of consolidation of 
a stationary suspension. Stationarity here implies little horizontal 
movement, although consolidation does mean that there is vertical 
(downward) movement. A stationary suspension, a partially consolidated 
bed and a settled bed may erode if the shear stress exceeds a certain 
critical value. Erosion of a stationary suspension is referred to as 
redispersion while erosion of a partially consolidated as well as a 
settled bed is termed resuspension. 



45 




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05 



CHAPTER III 
SEDIMENT TRANSPORT MECHANICS 

3.1 Introductory Note 

The purpose of this chapter is 1) to discuss the mechanics of 
cohesive sediment transport and 2) to describe the algorithms developed 
during this investigation. The processes for which algorithms have been 
developed include erosion, dispersive transport, deposition, bed 
formation and subsequent consolidation. The chapter begins with a 
description of the equations which govern the depth-averaged, uncoupled 
movement of water and suspended cohesive sediments. 

3.2 Governing Equations 

3.2.1 Coordinate System 

A right-handed Cartesian coordinate system is used (Fig. 3.1). The 
positive X-axis is coincident with the longitudinal axis of the estuary 
and points downstream. The coordinate system origin is located at some 
datum below the bed level. The positive z-axis is the vertical 
dimension and points upward. The y-axis defines lateral distances and 
points from right to left. 

3.2.2 Equations of Motion 

The equations which govern the two-dimensional, depth-averaged 

unsteady turbulent movement of an incompressible viscous fluid are 

statements which express two of the basic principles of Newtonian 

47 



48 




49 

physics, that of conservation of mass (continuity equation) and the 
conservation of momentum (equations of motion). These equations are 
solved numerically in order to describe the velocity field in the 
estuary or other water body of interest. Alternately, the velocity 
field may be measured in a physical scale model of the estuary. 
However, in this study of the "uncoupled" movement of cohesive 
sediments, these governing equations are not solved as it is assumed 
that the velocity field is known beforehand. The continuity equation 
and the two equations of motion are included and discussed here for the 
sake of completeness. 

3.2.2.1. Continuity 

The conservation of mass, as expressed by the continuity equation, 
states that the mass of an incompressible fluid entering a control 
volume per unit time is equal to the sum of the fluid mass leaving the 
control volume plus the change in volume of the control volume. The 
depth-averaged continuity equation for an incompressible fluid is 

ad a a 

— +— (u.d)+ (v.d) = 0 (3.1) 

at ax ay 

where d = depth of flow and u, v = time and depth-averaged water 
velocity components in the x- and and y - directions, respectively. 

3.2.2.2. Conservation of Momemtum 

The conservation of momentum (which is Newton's second law of 
motion) for an incompressible fluid states that the product of the fluid 
mass and acceleration is equal to the sum of the body (gravitational) 



50 

forces and the normal (pressure) and tangent (friction) surface forces 
which act on the boundaries of the water body. The two-dimensional, 
depth-averaged equations of motion for an incompressible viscous fluid, 
which can be derived from the Navier-Stokes equations, are given by 



Qu 5li Qu 1 1 5 a 

— + u — + V — = + — [ — h — ^ 1 + 2wvsin<j) - 

at dx ay dx 5x ay 



(uSv^)/2+ _L_a cos(e) 

C^d p d 

w 



av av av i ap i a 5 

_+u— +v— = -+ — [— T + — T 1 - 2oJusin* + 

at ax ay p^ dy p^ dx ay 



(3.2) 



2 2I/0 Pa'^a 2 

— (u +v^)^2+^V%sin(e) (3.3) 



8d p d 

w 



az 



in which: 

p = pressure force 

P^^ = fluid density 

Pg = air density 

■^■jj = horizontal turbulent shear stresses 

w = angular velocity of the earth 

<t> = local latitude 



Ufa*- 



51 

g = acceleration due to gravity 

f = Darcy-Vieisbach friction factor 

Vg = wind speed at a reference elevation 

above the water surface 
0 = angle between the wind direction and the 

positive X-axis 
= wind drag coefficient 

Equation 3.4 is the hydrostatic equation which results when the vertical 
component of the flow velocity and acceleration are small relative to 
the horizontal flow velocity and acceleration. The third term on the 
right hand side of Eqs. 3.2 and 3.3 is the Coriolis acceleration in the 
Northern hemisphere in the x- and y- directions, respectively. The 
fourth and fifth terms on the right hand side of Eqs. 3.2 and 3.3 
represent the effects of bottom shear stresses and surface wind shear 
stresses in the x- and y- directions, respectively. The three terms on 
the left hand side of Eqs. 3.2 and 3.3 represent the substantive fluid 
acceleration in the x- and y- directions, respectively. 

3.2.3. Advection - Dispersion Equation 

The principle of conservation of mass with appropriate source and 
sink terms describes the advective and dispersive transport of suspended 
sediment in a turbulent flow field. In this law, expressed by the 
advection-dispersion equation, the time-rate of change of mass of 
sediment 1n a stationary control volume is equated to the spatial rate 
of change of mass due to advection by an external flow field plus the 
spatial rate of change of mass due to diffusion and dispersion 



52 



processes. Both the three-dimensional form and the two-dimensional, 
depth-averaged forms of the advection-dispersion equation are derived in 
Appendix A. The latter is given here: 



C = mass of sediment per unit volume of water 
and sediment mixture 

D^-j = effective sediment dispersion tensor 

Sj = source/sink term. 
Implicit in Eq. 3.5 is the assumption that the suspended material is 
advected in the x- and y- directions at the respective water velocity 
components. This assumption is reasonable for sediment that is not 
transported as bed load since rolling and saltation of the sediment, 
which occurs during bed load transport, can cause a significant 
difference between the water and sediment velocities. Sayre (1968) 
verified that this assumption is approximately true for sediment 
particles less than about 100 m in diameter. The source/sink term in 
this equation can be expressed as 



9 5 a d ac ac 

— (dC) + u — (dC) +v — (dC) = — {dD — + dD — } + 

at ax ay ax ^^ax ^^ay 



a ac ac 




I — + dD — } + 

y'^ax yyay ^ 



(3.5) 



where: 



dC dC 



S = (— I + —I 
' dt^ dt'^ 



(3.6) 



L 



53 

dC, 

where — L is the rate of sediment addition (i.e. source) due to erosion 

dC 

from the bed, and —l^j is the rate of sediment removal (i.e., sink) due 

. . dC, dC 

to deposition of sediment. Expressions for — L and — L are qiven 

dt ^ dt^ 

respectively in Sections 3.4.3 and 3.6.3. S[_ accounts for the removal 
(sink) of a certain mass of sediment, for example, by dredging in one 
area (e.g. navagational channel) of a water body, and the dumping 
(source) of the sediment as dredge spoil in another location in the same 
body of water. 

In the following section, the schematization for sediment beds is 
described. This description is preceded by a general discusson on the 
nature (i.e. structure) of these beds as revealed in several laboratory 
investigations. 

3.3 Sediment Bed 
3.3.1 Bed Structure 

Surficial layers of estuarial beds, typically composed of flow- 
deposited cohesive sediments, occur in three different states: 
stationary suspensions, partially consolidated (or consolidating) beds 
and settled (or fully consolidated) beds. Stationary suspensions are 
defined by Parker and Lee (1979) as assemblages of high concentrations 
of sediment particles that are supported jointly by the water and the 
developing skeletal soil framework, and which have no horizontal 
movement. These suspensions, which may be regarded as extremely under 

consolidated soil, develop whenever the settling rate of concentrated 
mobile suspensions exceeds the rate of self-weight consolidation (Parker 

and Kirby, 1982). They tend to have a high water content (therefore low 

bulk density) and a very low, but measurable, shear strength, v, ^^^^ 



54 

must be at least as high as the bed shear, -v^, which existed during the 
deposition period (Mehta et al_., 1982a). Thus, they exhibit a definite 
non-Newtonian rheology. Kirby and Parker (1977) found that stationary 
suspensions have a surface bulk density of approximately 1050 kg/m^ and 
a layered structure. Krone (1963) found that, in addition to the bed 
shear, the structure (or framework) of these suspensions depends on the 
aggregate order in the following manner: if the aggregates deposit 
without being broken up by the bed shear, the surficial layers of these 
suspensions will be composed of an aggregate network whose order is one 
higher than that of the individual settling aggregates; therefore, these 
layers will have lower bulk densities and shear strengths than those of 
the aggregates which form them. 

Whether or not entrainment of these suspensions, also referred to 
as redispersion (Parker and Kirby, 1977) and mass erosion (Paaswell, 
1973), occurs during periods of erosion depends upon the mechanical 
shear strength (i.e. stability) of this aggregate network. That portion 
which remains on the bed undergoes: 1) self-weight consolidation, due 
to overburden pressure resulting from the weight of the overlying 
sediment which crushes the aggregate network below, and 2) thixotropic 
effects, defined as the slow rearrangement of deposited aggregates 
attributed to internal energy and unbalanced internal stresses 
(r^itchell, 1961), both of which reduce the order of aggregation of the 
sub-surface bed layers. This implies that the bed becomes stratified 
with respect to bulk density and shear strength, with both properties 
typically increasing montonically with depth, at least under laboratory 
conditions (Mehta et ail_., 1982a). Stationary suspensions generally have 
a lifespan that varies from a few hours to a few days. Differential 



55 



settling caused by sorting processes is another cause of stratified bed 
formation. 

Continued consolidation eventually results in the formation of 
settled mud, defined by Parker and Lee (1979) as "assemblages of 
particles predominantly supported by the effective contact stresses 
between particles as well as any excess pore water pressure." This 
portion of the bed has a lower water content, a lower order of 
aggregation, and a higher shear strength and therefore is better able to 
resist high bed shear stresses. The settled mud in the Severn Estuary 
and Inner Bristol Channel, United Kingdom, has a bulk density range from 
1,300 to 1,700 kg/m^ (Kirby and Parker, 1983). 

In this study the primary characteristic used to distinguish 
between a stationary suspension and a partially consolidated or settled 
bed is the mode of failure that occurs when the surface of the 
suspension or bed is subjected to an excess shear stress (i.e. i^i^ > 
t;^.). Erosion by particle by particle or aggregate by aggregate removal 
is not a correct representation in areas where stationary suspensions 
exist (Kirby and Parker, 1983). As stated previously, stationary 
suspensions undergo redispersion while partially consolidated and/or 
settled beds undergo resuspension (Parker and Kirby, 1977) or surface 
erosion (Paaswell, 1973). Both erosion processes are discussed in 
Section 3.4.1. 

The nature of the density and shear strength profiles typically 
found in flow-deposited cohesive sediment beds has been revealed in 
laboratory tests by, among others, Richards _et_^. (1974), Owen (1975), 
Thorn and Parsons (1980), Parchure (1980), Bain (1981) and Dixit 
(1982). A review of this subject is given here. 



56 

Figure 3.2 shows the dimensionless density profile measured by Owen 
(1970) after 4.2 and 8.3 hours of consolidation for mud obtained from 
the Thames near Dagenham, England. The indeterminate effect of salinity 
on the density profile for two different beds after 67 hours of 
consolidation for mud obtained at the entrance to the Royal Edwards 
Docks, Avonmouth, Bristol, England is seen in Fig. 3.3. In fact, 
despite the difference in mean density the same dimensionless profile is 
drawn through the data points in both Figures 3.3(a) and 3.3(b). The 
only distinguishable difference between these two sets of density 
profiles is the slightly lower relative surface densities in the bed 
formed by settling of the lower initial suspension concentration of 7.72 
g/1. The Avonmouth mud is composed predominantly of illite, which is a 
relatively inert clay mineral, and was found to have a CEC value of 17 
meq/100 gm. Therefore, it is not surprising that salinity had very 
little effect on the bed density. 

Figure 3.4 shows the measured dimensionless density profiles for 
four different bed thicknesses after 67 hours of consolidation for the 
Avonmouth mud (Owen, 1970). The same average profile drawn in Figure 
3.3 was drawn on this figure as well. A remarkable fit obtained between 
the data and this common dimensionless profile is evident. 

Figure 3.5 shows the dimensionless density profiles obtained by 
Thorn and Parsons (1980) after two days of consolidation for muds from 
the Forth Estuary at Grangemouth, Scotland, the Brisbane River at the 
Port of Brisbane, Australia, and the dredged channel to the Port of 
Belawan, Sumatra, Indonesia. The percentage of clay minerals, 
percentage of non-clay minerals and the cation exchange capacity for the 
Grangemough mud, Brisbane mud and Belawan mud were 51%, 50% and 75-80% 



57 




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58 




59 




o 



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60 

(clay minerals), 39%, 50% and 20% (non-clays) and 20 meg/100 gm, 35 
meg/100 gm and 25 meg/100 gm (CEC), respectively. Thus, the Grangemouth 
mud is the least cohesive and the Brisbane mud is the most cohesive. 
This is not unexpected as the Brisbane mud has the highest percentage of 
montmorillonite (approximately 60% of the clay mineral fraction), which 
is a very active (cohesive) clay mineral. 

Figure 3.6 shows the dimensionless density profiles (normalized 
with respect to the initial suspension concentration) measured by 
Parchure (1980) for commercial grade kaolinite after 24, 40 and 135 
hours of consolidation. This clay had a CEC of approximately 9 
meq/100 gm. 

Figure 3.7 shows the dimensionless density profile for mud from the 
Mersey Estuary, England after 48 hours of consolidation (Bain, 1981). 
This mud was composed of 76% clay minerals and 24% silica. 

Figure 3.8 shows dimensionless density profiles of two different 
beds of the same natural mud after 48 hours of consolidation (Thorn, 
1981). The solid line profile was measured using a nuclear 
transmissometer, while the discrete point profile was determined using a 
layer-by-layer sampling technique. 

Figure 3.9 shows the dimensionless density profiles found by Dixit 
(1982) for flow-deposited beds of commercial grade kaolinite after 2, 5, 
11 and 24 hours of consolidation while Figure 3.10 shows these profiles 
after 48, 72, 96, 144 and 240 hours of consolidation. 

It is evident from all these density profiles that a static or 
dynamic deposited cohesive sediment bed has 1) a characteristic 
elongated^ slightly reversed S shape density profile that generally 
increases monotonically with depth and that is independent of the bed 



61 




Fig. 3.6. Variation of Bed Density v/ith Depth for Three Different 
Conditions of Flow Deposited Beds (after Parchure, 1980). 



62 




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63 





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H/(^z-H 



54 



thickness and 2) a very Tow, generally indeterminate surface density. 
The variation of the density profile with consolidation time is examined 
in Chapter III, Section 3.7. 

Parchure (1980) made the following observation with regard to the 
shear strength profiles, in flow-deposited cohesive sediment 

beds: '^^(z) increases rapidly with distance below the water-sediment 
interface for z < i^^, at which i^ = t^j^, where t^i^ (defined in Section 
3.4.1) is a characteristic value of -r^. For z > z^^, ^^^(z) continues to 
increase but at a greatly decreased rate (Figure 3.11). The influence 
of such a '^(-iz) profile on the erosion rate is discussed in the 
following section. Figure 3.12 shows a '^(^iz) profile found by Dixit 
(1982). From the tests conducted by Dixit, the following two 
observations may be made: 1) such a "^(.iz) profile was not found in five 
out of nine experiments, and 2) the sediment beds used by Dixit were up 
to six times thicker than those used by Parchure. Therefore, the t^(.(z) 
profiles measured by Dixit are naturally more representative of 
estuarine beds, and as such it is believed that until further studies 
are conducted, no definitive statement regarding the precise nature of 
"^(.{z) profiles in cohesive sediment beds can be made. The possibility 
of a correlation between the bed density and shear strength of cohesive 
sediment beds is examined in Section 3.7. 

3.3.2. Effect of Salinity on Bed Structure 

For most cohesive soils the interparticle and interfloc contact is 
considered to be the only significant region between particles where 
normal stresses and shear stresses can be transmitted (Mitchell et al.. 



65 



E 
£ 



u 
o 
< 

Li. 

tr 

UJ 
H 




Expt. 17 

=0.05 H/m 

T^,= 24 hrs 
Tch = 0.2l N/m^ 



hJ 
< 

I 

I- 
u 



o 

UJ 

cn 

$ 

O 
-J 
LU 
OQ 

X 

o. 

Q 




Expt. 18 
Tb =0.015 N/m^ 
V 40 hrs 

T. =0.28 N/m' 
cn 



Expt. 19 

Tt, = 0 N/m' 

T^= I35tirs 



T^h=0.34N/m 



Fig. 3.11. 



BED SHEAR STRENGTH (N/m^) 

Bed Shear Strength Profiles for Kaolinite Beds 
(after Parchure, 1980). 



, — — — 




Fig, 3.12. Bed Shear Strength Profile for a Kaolinite Bed (after 
Dixit, 1982). 



67 

1969). In particular, it seems very likely that the primary role of the 
double-layer interaction and other physicochemi cal forces is to control 
the structure of the soil and to alter the transmitted stresses from 
what they would be due to the applied flow-induced shear and overburden 
normal stresses alone. Two factors that effect the structure of a 
cohesive soil, swelling and permeability, and the effect salinity has on 
these factors are discussed next. 

The degree of swelling which occurs when a soil is immersed in a 
fluid is Influenced by factors such as the amount of clay, shape and 
size of the particles, the salinity and the sodium adsorption ratio 
(SAR) of the eroding and pore fluid, and the presence of an imposed load 
on the swelling areas (Grimshaw, 1971). Sargunam et aK (1973) state 
that decreases in the salinity of the eroding fluid or increases in the 
SAR cause the surface clay particles to swell more. This swelling 
causes a weakening of the interparticle attractive forces and thus 
increases the susceptibility of the soil to erosion. Increasing the 
salinity of the eroding fluid causes a greater compression of the 
diffuse layer, thereby reducing the repulsive forces of soil 
particles,. This reduction serves to limit the amount of swelling. 

Sargunam et_^. (1973) found that when the salinity of the eroding 
fluid is greater than that of the pore fluid, the yield strength of the 
soil is greater and therefore the erosion potential is decreased. In 
this case the osmotic pressure gradient across the fluid-bed interface 
may result in deswelling, or consolidation of the bed sediment 
particles, which would cause an increase in the interparticle bonding 
forces and therefore lessen the susceptibility to erosion. 



68 

It 13 believed that while this phenomenon of swelling influences to 
some degree the structure and hence the erosion potential of a cohesive 
bed, it is not nearly as significant as the upward flux of pore water 
due to gravitational forces in a consolidating mud. 

Quirk and Schofield (1955) found that the degree of permeability of 
clay soils depends upon the nature and the concentration of the cations 
present in both the eroding and pore fluids. In particular they found 
that permeability increased with an increase in the salinity of the 
eroding fluid. Swelling, stabilization (i.e. decoagulation) and 
consolidation are generally considered to be the main reasons for 
changes in the permeability. The former can cause either partial or 
total blockage of soil pores which would result in a decrease in 
permeability. Stability essentially occurs during swelling when the 
clay particles have separated to the extent that the interparticle 
repulsive forces are dominant over the attractive forces. Since an 
increase 1n the salinity of the eroding fluid serves to limit the amount 
of swelling which occurs and thus restricts the amount of stabilization 
as well, an increase in salinity would result in increased 
permeability. The converse was found to occur as well since, as stated 
previously, a decrease in the salinity of the eroding fluid causes an 
increase in the degree of swelling and stabilization. 

As mentioned previously, the effect the salinity of the pore fluid 
has on the bed density could be expected to be a direct function of the 
cation exchange capacity of the sediment. Salinity was seen (in Figure 
3.3) to have very little effect on the density profile for the 
relatively inert Avonmouth mud. Figure 3.13 shows the indeterminate 
effect of salinity on the bed density profile for mud from Lake Francis, 



69 




70 

Nebraska, of which 50% was finer than 2 m (clay-sized particles), with 
montmorinonite, illite, kaolinite and quartz being the predominant 
minerals, and with a CEC of 100 meq/ion gm. This high CEC value 
indicates a higher percentage of montmorillonite than the other two clay 
minerals. Evidently, salinity and the CEC value had very little effect 
on the bed density profile. 

Using the method described, by Mehta et_al_. (1982a), the bed shear 
strength (or the critical shear stress for erosion), t^, of Lake Francis 
mud as a function of depth below the initial bed surface, z^, was 
determined as a function of salinity for salinities from 0 to 10 ppt 
(Figure 3.14). Two trends are observed in this graph. First, 
increases with depth in the upper part of the bed for all salinities (no 
definite data could be obtained for the lower part of the bed, i.e. for 




0-0 0.2 0.4 0.6 0.8 

BED SHEAR STRENGTH, (N/m^) 
Fig, 3.14. Bed Shear Strength Profiles as Functions of Salinity. 



71 

> 0.5 cm, inasmuch as this portion of the bed did not erode during 
these experiments). Second, increases with increasing salinity from 
0 to 2 ppt; thereafter, for salinities up to 10 ppt, no measurable 
increase in '^^ occurred. 

3.3.3. Bed Schematization 

To facilitate the modeling of changes in the bed surface elevation 
due to erosion, deposition and consolidation processes, the bed is 
treated in the following manner: 1) it is discretized into a number of 
layers and 2) the bed properties, e.g. thickness, are assumed to be 
spatially (in the x-y plane) invariant within each element, but not so 
from element to element, in order to account for inter-element spatial 
variances in shoaling and/or scouring patterns. These two factors are 
expounded upon below. 

The bed in each element is considered to be composed of two 
sections: 1) the original, settled (consolidated) bed that is present 
at the start of modeling and 2) new deposits located on top of the 
original bed, that result from deposition during the modeling. Each of 
these two sections is divided into a number of layers in order to 
specify the actual shear strength and bulk density profiles in the 
model. The new deposit bed section is subdivided into two sub-sections, 
the top referred to as unconsolidated new deposit (UNO) layers and the 
bottom as partially consolidated new deposit (CND) layers (Figure 
3.15). The former sub-section, i.e. the one corresponding to a 
stationary suspension, is considered to undergo redispersion while the 
latter, i.e. the partially consolidated bed, undergoes resuspension when 
subjected to an excess shear stress. The settled bed as well undergoes 



72 



© 



Unconsolidated 
New Deposit 
(UND) 



Partially (2) 

Consolidated ^ 
New Deposit 

(PCND) ^ 



® 



® 



® 



Tu 



Bed Surface 



TuNDJ^TLAYMd)) 

^ ' 



T 



UND, 2 



NLAYTM = 3 



T, 



UND, 3 



TcND.i (TLAY(I)) 



New Deposits 



CND,2 



CND, 3 



NLAYT=4 



T, 



CND, 4 



To, I (THICKO(I)) 



T 



0,2 



Settled Bed 



T 



0,3 



NLAY0=4 



To,4 



Fig. 3.15. Bed Schematization used in Bed Fonnation Algorithm. 



Fig. 3 



BED SHEAR STRENGTH , ( N/m^) 
0.2 




16. 



Hypothetical Shear Strength Profile Illustrating 
Determination of Bed Layers Thicknesses. 



73 

resuspension. The number of layers indicated in Fig. 3.15 for each of 
the three bed sections are not fixed, as each section can be assigned 
any given number of layers. 

Stationary suspensions are represented in the depth-averaged model 
as being the top section of the layered bed model, even though they are 
not a true bed or soil, in order to account for the subsequent 
redispersion and/or consolidation of these suspensions. However, the 
time varying thickness of the bed in each element is equal to the sum of 
only the NLAYT CNO layers and the NLAYO settled bed layers. The 
following bed-related parameters are required for bed schematization in 
the model : 

1) The bed shear strength profile in the UND. This can be ascertained 
from laboratory erosion tests using samples of the sediment from 
the water body being modeled (see Appendix D, Section D.2). 

2) The number of UND layers (NLAYTM) and the thickness of each layer 
(TLAYM(I), 1=1, MLAYTM). These parameters must be determined using 
the shear strength profile. For example. Figure 3.16 shows a 
hypothetical t^(z) profile and illustrates that this bed section 
must be divided such that the variation of within each layer is 
approximately linear. The values at the NLAYTM+1 nodes need to 
be read into the model. 

3) The dry sediment density values at the NLAYTM+1 nodes as well need 
to be determined. The p(z) profile may be determined using a 
laboratory freeze-dryi ng method (appropriately modified for field 
samples, where necessary) described by Parchure (1980), the pumping 
method described by Thorn and Parsons (1977), a gamma-ray nuclear 
transmission densitometer (Whitmarsh, 1971) or a nondestructive X- 
ray technique (Been and Sills, 1981). 



74 

4) The same parameters for the CND layers and the settled bed layers 
must be determined, with the settled bed parameters determined for 
each element (where an original bed exists). The parameters read 
in for the UND and CND layers are assumed to be constants for all 
elements. 

5) A stationary suspension and/or partially consolidated bed present 
on top of the settled bed at the start of the modeling is simulated 
by reading in the dry sediment mass per unit bed area of such new 
deposits obtained from every element where such exist. 

The bed level at which the dry sediment density is approximately 
480 kg/m^ is usually taken to be the top of the settled bed. Thus, the 
sediment located above this level is considered to be new deposits. 
Another method which may be used to differentiate between new deposits 
and the settled bed is described in Appendix D, Section D.2. 

Included in Appendix D is a brief description of how these various 
bed properties can be determined through the use of a field data 
collection program and a laboratory testing program. Other parameters 
characterizing the rate of resuspension that each layer undergoes when 
subjected to an excess shear must be evaluated as well; these are 
discussed in Section 3.4.3. 

The following procedure was developed for forming the new deposit 
bed layer(s) which result from new deposits initially present on top of 
the settled bed and/or deposited during the modeling. 

The dry sediment mass per unit bed area per element, Mg, read in 
initially if a new deposit exists in any given element, or deposited 
during modeling (as determined by the deposition algorithm) is used in 
conjunction with the UND and CND properties to solve iteratively for the 



75 

thickness of bed formed by Mq for each element where Mq > 0. This 
thickness, at, depends on the dry sediment density profile, p(z), for 
the UND and CND layers. The thickness AT is determined using the 
following relationship: 

J* P(2)zdz 

AT = ~ (3.7) 

where * = AT ± 0.02AT. If AT is greater than TLAYM(l) (see Figure 3.15) 
then more than one layer of UND is added. The assumed linear variation 
of p within each layer is used in the above equation. When or if the 
UND layers are filled, the same procedure is used to fill up the CND 
layers below the UND layers. The bottom CND layer can never fill up; 
therefore, continuing deposition is accounted for by increasing the 
thickness of this layer, while the thicknesses of the overlying UND and 
CND layers remain the same. This particular filling sequence was used 
in order to account for the consolidation of the sediment bed due to 
overburden pressure during the bed formation phase by virtue of the 
increasing and p values with bed depth. 

In the next section, a discussion on the erosive behavior of fine 
sediment beds is given, followed by a description of the erosion 
algorithm. 

3.4 Erosion 

3.4.1. Previous Investigations 

Of interest in this study are the erosion (resuspension) 
characteristics of saturated, flow-deposited cohesive sediment beds. A 
number of laboratory investigations were carried out in the sixties and 



76 

early seventies in order to determine the rate of resuspension, e, 
defined as the mass of sediment eroded per unit bed surface area per 
unit time, as a function of a bed shear stress in steady, turbulent 
flows. An important conclusion from these tests was that the usual soil 
indices such as liquid or plastic limit do not adequately describe the 
erosive behavior of these soils (Mehta, 1981). For example, 
Partheniades (1962) concluded that the bed shear strength as measured by 
standard tests, e.g. the direct-shear test (Terzaghi and Peck, 1960), 
has no direct relationship with the soil's resistance to erosion, which 
is essentially governed by the strength of the interparticle and inter- 
aggregate bonds between the deposited sediment material. Shown in Table 
3.1 are various physicochemical factors known to govern the erosive 
properties of these beds. These factors must be specified to properly 
characterize the erosive behavior. The hydrodynamic factors define the 
erosive forces while the bed and fluid physicochemical properties 
determine the resistivity of the bed to erosion. 

The erosive forces, characterized by the flow-induced instantaneous 
bed shear stress, are determined by the flow characteristics and the 
surface roughness of the fluid-bed interface. The sediment composition, 
pore and eroding fluid compositions and the structure of the flow- 
deposited bed at the onset of erosion must be determined in order to 
properly define the erosion resistance of the bed. Sediment composition 
is specified by the grain size distribution of the bed material (i.e. 
weight fraction of clays, silts), the type of clay minerals present, and 
the amount and type of organic matter. The CEC can be used to 
characterize clay composition using the apparent dielectric constant 
measured at selected frequencies. Each clay tested appeared to have a 



77 



Table 3.1 

Principle Factors Controlling Erosion of 
Saturated Cohesive Sediment Beds 



HYDRODYNAMIC FACTORS (Erosive Force) 



BED SHEAR STRESS 



'Flow Characteristics 



•Bed - Fluid Interface 



BED AND FLUID PROPERTIES (Resistive Force) 



SEDIMENT COMPOSITION 



PORE FLUID 
COMPOSITION 



ERODING FLUID 
COMPOSITION 



■Clay Mineral Type] Ion Exchange Capacity 
•Clay Percentage by Weight 
■Organic Matter 



■ Mono-and Divalent Cations Concentrations ] Conductivity 
Relative Abundance of \ car l(Na"*",Ca"^.Mg^) 
Mono-and Divalent Cations/ ^'^'^ v. , . y 

Temperature 
pH 



Salinity (NaCI,CaCl2,MgCl2) 
■Temperature 

•pH 

Cementing Agents (Iron Oxide, etc) 



BED STRUCTURE 



/Placed Bed 
■Stress History I D3pQ3i^3d 



78 

characteristic value of a "dielectric dispersion" parameter determined 
from these measurements. The dielectric dispersion has been defined as 
the amount of decrease in the apparent dielectric constant with 
frequency (Alizadeh, 1974). 

The composition of the pore and eroding fluid are specified by the 
temperature, pH, total amount of salts and the type and abundance of 
ions present, principally CI", Na+, Ca^^, and Mg2+. Cementing agents 
such as iron oxide can significantly increase the resistance of a 
sediment bed to erosion. Measurement of the electrical conductivity is 
used to determine the total salt concentration. The physicochemical 
aspects pertaining to the aforementioned factors have been summarized by 
Sargunam et _al_. (1973), Kandiah (1974), Arulanandan et_ al_. (1975) and 
Ariathurai and Arulanandan (1978). The effect of the bed structure, 
specifically the vertical dry sediment density and shear strength 
profiles, on the rate of erosion is discussed by Lambermont and Lebon 
(1978) and f^ehta et al_. (1982a). 

Several different types of relationships between the rate of 
erosion, £, and the time-mean value of the flow-induced bed shear 
stress, Xj^, have been reported for non-stratified beds. These include 
statistical-mechanical models (Partheniades, 1965; Christensen, 1965), a 
rate process model (Paaswell, 1973; Kelly and Gularte, 1981) and 
empirical relationships (Ariathurai and Arulanandan, 1978). These 
relationships typically have the following general form (Mehta, 1981): 



(3.8) 



where ili,^!^, . . . ,ti^. are parameters that specify the bed resistivity. 



79 

The resuspension rate, £, is related to the time-rate of change of 
the suspension concentration, dC/dt, and to the time-rate of change of 
the depth of erosion, Zj^, with respect to the original bed surface 
elevation, by the following expressions: 



dC 

e = d— (3.9) 
dt 



dC 1 dz. 

— = -p(z.)— (3.10) 
dt d ° dt 



Figure 3.17 shows the general nature of laboratory determined 
relationships found by, among others, Partheniades (1962) and 
Christensen and Das (1973) for placed beds. Placed beds are sediment 
beds that are formed artificially. Such beds include those that are 
remolded and/or compacted after placement in a test apparatus (Mehta and 
Partheniades, 1979). In these beds, the shear strength and density 
profiles show much less significant stratification over the depth of the 
bed than in flow-deposited beds (Mehta et_ al_. , 1982a). Shown in Figs. 
3.18 and 3.19 are examples of this relationship, which may be expressed as 



^ = ^^^V^ch^ ^ ^ch ^'-''^ 



80 




Bed shear stress 

Fig. 3.17. Laboratory Determined Relationship Between Erosion Rate, e, 
and Bed Shear Stress, (after Mehta, 1981). 

^ I 1 1 I 1 i 1 1 1 1 1 




Fig. 3.18. Example of Relationship Betv/een e and x, (after Mehta, 1981). 



81 




Fig, 3.19. e - T^t) ^^^^ °^ Partheniades (1962), Series I and II 
(after Mehta, 1981) . 



82 

where 'A = slope, and the subscript ch refers to a characteristic value 
(flehta, 1981; Hunt, 1981). For x^ < x^^, M = and for t > t^^, M = 
Mg. Thus, Eq. 3.11 has the general form z = e {x^, x^^, z^^, M^, 
in Eq. 3.8. The parameter x^^^ determined by extrapolation of the M2 
line to the £ = 0 axis, has been interpreted to be the critical shear 
stress for erosion (Partheniades, 1962; Gularte, 1978). The 
characteristic shear strength, t^^^ is defined to be the value of the 
bed shear stress at which the M^^ and M2 lines intersect. Values of s^.^, 
'''ch' '^1 ^^2 largely determined by the physicochemical factors 
given in Table 3.1. 

Ariathurai and Arulanandan (1978) found the same general 
relationship for remolded beds as given in Eq. 3.11, but with = M^. 
Thus, Eq. 3.11 becomes 



b cr 



£ = M (- 



cr 



(3.12) 



where M = M't^^^. Figure 3.20 gives an example of this relationship, 
with ot^ = 1/m'. 

Figure 3.21 shows the measured variation of C with time typically 
found by several investigators (Partheniades, 1962; Mehta and 
Partheniades, 1979; Mehta et_ al_. , 1982a) in laboratory resuspension 
tests with flow-deposited (stratified) beds under a constant applied 
■^5. As observed, dC/dt is high initially, decreases monotonically with 
time and appears to approach zero. The value of x^ at the depth of 
erosion at which dC/dt, and therefore e becomes essentially zero has 



83 



2.0 



't5 



Ariathurai and Arulanandan ( 1978) 
30% illite 




20 



3.20. Dimensionless e - t Relationship Based on Results of 
Ariathurai and Arulanandan (1978) (after Mehta, 1981) 




-1x1 



2 4 6 8 10 12 14 16 18 20 22 24 
TIME (Hours) 



3.21. Relative Suspended Sediment Concentration Versus Time for 
a Stratified Bed (after Mehta and Partheniades , 1979). 



84 

been interpreted to be equal to (Mehta et aT_. , 1982a). This 
interpretation is based on the hypothesis that erosion continues as long 
as \ > T^, i.e. the excess shear stress x^-t^ > o. Erosion is arrested 
at the bed level at which x^-x^ = o. This interpretation, coupled with 
measurement of p(z^) and the variation of C with t can result in an 
empirical relationship for the rate of erosion of stratified beds. 

Resuspension experiments with deposited (stratified) beds were 
performed by Parchure (1980) in a rotating annular flume and by Dixit 
(1982) in a recirculating straight flume. Both flumes are described in 
Chapter V, Section 5.2. The objective of these experiments was to 
determine the effect of varying bed shear strength with depth below the 
initial bed surface on the rate of surface erosion under a flow-induced 
shear stress. A description of the experimental procedures and results 
from these experiments have been given by Parchure (1980), Dixit (1982) 
and Mehta etal_. (1982a). A synopsis is given here. 

A commercial grade kaolinite with a CEC of approximately 9 meq/100 
gm was used in these experiments. Tap water, with a total salt 
concentration of 0.28 ppt, pH = 8.5 and sodium adsorption ratio SAR = 
0.012, was used in the recirculating flume, while tap water plus 
commercially available sodium chloride at 35 ppt concentration, pH = 8.1 
and SAR = 12.0 was used in the annular flume. The kaolinite was 
equilibrated with the fluid for at least two weeks prior to the tests. 
The equilibration time is an important factor that can affect the rate 
of erosion due to the possibility of concentration gradients of ionic 
constituents between the solid and the liquid phases, or between the 
pore fluid and the eroding fluid, in the sediment-water system if the 
time allowed for equilibration is insufficient (Mehta, 1981). 



85 

The resuspension test methodology is depicted in Fig. 3.22. 

Specifically, this figure shows how the bed shear was varied over the 

course of each experiment. In Phase I the sediment-water mixture with 

sediment concentration was mixed at a high shear stress, i;^, for a 

period T^. The shear stress t was greater than v , the maximum bed 

max 

shear stress at which deposition of suspended sediment occurs. In Phase 

II the bed shear was lowered in steps, to for T^ , then to for 

Tj^ and finally to zero shear stress for a period of consolidation, 

J^^. During this phase the sediment settled out of suspension, formed a 

bed and began to consolidate. As indicated in Fig. 3.22, the first two 

phases define the pre-erosion stress history of the bed. In Phase III, 

the shear stress was increased as shown in discretized (one-hour) steps, 

i.e. = T2 = ...T^. (= 1 hour), and resuspension of the deposited 

material occurred. 

The following parameters were held constant in most of the 

experiments in each flume: C^, x^. T^, t^^, t^^, t^^' ^nd T^- . 

Shown in Fig. 3.23 are typical values for these parameters and the 

measured variation of suspended sediment concentration with time for a 

test in tap water. The parameter At^. , which may be referred to as the 

normalized incremental bed shear stress, is defined as (x. ^.-xAf-z. 

1+1 1 1 

where t. is the bed shear stress, -z^, during the ith time- step. 

In steps i =1,...,5 in this figure it is apparent that the 
suspension concentration approaches a constant value during the latter 
stages of each time-step, i.e. dC/dt^, while for steps i = 5 and 
especially 7 the values of dC/dt do not approach zero over the one hour 
periods. This difference in the concentration-time profiles is 
represented in a different manner in Fig. 3.24, which shows the 



86 



0) 
A 



8? 



T 

i 

rO 

if 

hr 
_L 



CVJ 



CVJ 



a 

Jl 



1 



c 
g 

c 

CD 
Q. 
CO 
Z3 
CO 
<D 

IT 



V- 

O 



CO 
CD 



+ 1 



g 

(/) 
o 

LU 
I 

CD 

Q_ 



13 

CO • 

"O fO 
O) CM 
•r- CO 
I— CTl 
Q-r-l 

+J 

O 

fd 

o ^ 
•r- OI 

+-> s: 
-I- i. 

(O +-> 

fO 

QJ 

-C 

-M 
t. (/) 
O O) 
4- 1— 

>> C 
O 

O -r- 

I (/I 

o c 

-a oj 

o Q. 
to 

+j zs 
O) oo 
2: cu 

o; 

-a 

OJ -a 
4-) c 
o ra 
CD 

r- c 

o 



00 
4-5 

O) ra 
J= S- 

+J ra 

Q. 

q- QJ 
o s- 

c 

O T3 
•.- O) 

+J CQ 
n3 

4- > CD 
C C 
<D •!- 
to S- 
CU 3 

5- Q 
Q. 

O; to 
a: to 

O) 
CJ s- 
■1- +-> 

e s- 

CU ro 
^ OJ 

O J= 
00 00 



CM 
CVJ 

ro 

•I— 

a. 



87 



\ 
E 

CP 



o 

o 

< 
cr 

H 

u 
o 



o 

00 

z: 
a. 

CO 
Z) 
CO 



3.50 

11.00 



10.00 -Tr 



ELAPSED TIME T (hrs) 
4.50 5.50 6.50 



7.50 



6.00- 




0.00 



'•00 2.00 3.00 

ELAPSED TIME T (hrs) 



4.00 



Fig. 3.23. Variation of Suspension Concentration with Time for T =48 
Hours (after Dixit, 1982). dc 



88 



- CQ = 44.i gm/Iiter 
h =30.5 cm 




I 



Fig. 3.24. C(T^) Versus t^. for Three Values of T^^, Using Kaolinite 
in Salt Water (after Mehta et al. , 1982a). 



89 

suspension concentration at the end of step i, C(T^. ), plotted against t. 
for three different tests in the recirculating flume. The value of -v^^, 
a characteristic shear stress (similar to one defined previously for 
tests with placed beds) is determined as shown for each test. It is 
apparent that dC(T^-)/dt is higher for t^- > t^^^ than for i^ < x^^. The 
significance of this observation is better appreciated when it is 
realized that £ is proportional to the excess shear stress, and 
that increases more rapidly with depth, z^, for z^^ < z^,^, where z^^ 
is the depth below the initial bed surface at which = (Mehta et 
al_., 1982a). 

The following empirical relationship between and ■t:. = -v^{z^) was 
derived from these experiments: 



e. = ^,_^exp[a.. 



1 c b 



c b 



(3.13) 



where and <x. are empirical coefficients. Figures 3.25 and 3.26 show 

this relationship for tests in tap water and salt water, respectively. 

This relationship is analogous to the rate expression which results from 

a heuristic interpretation of the rate process theory for chemical 

reactions (Mehta et_ al_., 1982a). Christensen and Das (1973), Paaswell 

(1973) and Kelly and Gularte (1981) have used the rate process theory in 

explaining the erosional behavior of cohesive sediment beds. By 

analogy, e^- is a quantitative measure of the work done by t^- on the 

system, i.e. the bed, and e„ and a./i t-, \ ... ^ , 

' 0^- '^■\/ '-(.[z^] are measures of the system s 

internal energy, i.e. bed resistance to an applied external force. 



90 




Fig. 3.25. Normalized Rate of Erosion, e./e.^. Versus Normalized Excess 
Shear Stress, , Using Kaolinite in Tap 

Water (after Mehta et a^. , 1982a). 



91 




Series 3 
T(j(.= 40 hr 



i 






(g cmViin"') 


• 1 


0.100 


5.9 


0.04 


o 2 


0.120 


5.5 


0.25 


^ 3 


0.145 


5.5 


0.30 


a 4 


0.175 


5.5 


0.27 


* 5 


0.210 


84 


0.22 


1 1 1 1 


1.0 


i.5 


2.0 


2.5 



3.26. Normalized Rate of Erosion, e^/e^^. Versus Normalized 
Excess Shear Stress, (t^/t^U^)) /t^U^) , Using 
Kaolinite in Salt Water (after Mehta et al . , 1982a). 



92 

An important conclusion reached from the above experiments was that 
new deposits should be treated separately from settled, consolidated 
beds (Mehta et_ a1_. , 1982a). The rate of surface erosion of new deposits 
may be evaluated using Eq. 3.13, while the erosion rate for settled beds 
may be suitably determined using Eq. 3.12, in which £ varies linearly 
with the normalized excess bed shear stress. The reasons for this 
differentiation in determining e are twofold: 1) typical -^^ and p 
profiles in settled beds vary less significantly with depth than in new 
deposits, and may even be nearly invariant. Therefore, the value of 
^\/^c^ " ^ = ^"^b relatively' small . For small values of At,^, 

the exponential function in Eq. 3.13 can be approximated by a* (1 + Atj^) 
which represents the first two terms in the Taylor series expansion of 
exp{a(ATjj)). For small values of At^, i.e. L-z^ « i, both expressions 
for £ vary linearly with A-c^^. Thus, the variation of £ with depth in 
settled beds can be just as accurately and more simply determined using 
Eq. 3.12. 2) The laboratory resuspension tests (briefly described in 
Appendix D) required to evaluate the coefficients and a for each CND 
layer can not be practically or easily performed using vertical sections 
of the original settled bed (obtained from cores). A simpler laboratory 
test has been described by Ariathurai and Arulanandan (1978) to evaluate 
the variability of M with depth. This procedure is briefly noted in 
Appendix D. 

3.4.2. Effect of Salinity on Resuspension 

Sherard_et^. (1972) have shown that the susceptibility of a 
cohesive sediment bed to erosion depends on two factors: 1) the pore 
fluid composition, as characterized by the SAR, and 2) the salinity of 



93 

the eroding fluid. It was found that as the eroding fluid salinity 
decreases, soil resistivity to resuspension decreases as well. These 
results were verified by Arulanandan et al_. (1975). In addition, 
Sherard et a][. (1972) found that the erosion resistance decreased by 
either the exchange of cations or a reduction of the valence of the 
cations in the pore fluid. Kandiah (1974) and Arulanandan et al_. (1975) 
confirmed these findings by showing that the erosion resistance 
decreased and the rate of resuspension increased with increasing SAR 
(and therefore decreasing valency of the cations) of the pore fluid. 
Figure 3.27 shows such a relationship between the SAR and the critical 
shear stress for erosion, which is a measure of soil resistance to 
erosion (Alizadeh, 1974). 

Experiments were conducted during this study to determine the 
effect of the eroding fluid salinity on the rate of resuspension. The 
experiments were performed in the rotating annular flume using the 
bottom sediment from Lake Francis, Nebraska. Analysis of water from the 
lake indicated the presence of Na"^, K"^, Ca^"^, Mg^^, Al^"^, Fe^"*", CI" 
SO4 . These cations and anions would be expected to be present in the 
sediment as well. The average pH of the lake water was 8.6. The 
sediment was repeatedly washed in an attempt to remove these free salts 
so that their effect on the sediment properties was minimized. The 
washing was performed by immersing the sediment in deionized water, 
vigorously stirring the sediment and water, allowing time for the 
sediment to settle out of suspension by gravity, and then siphoning off 
the clear supernatant water. This procedure was repeated at least three 
times. 



94 




SODIUM ADSORPTION RATIO, SAR 



Fig. 3.27. Critical Shear Stress Versus SAR for Montmorillonitic 
Soil (after Alizadeh, 1974). 



95 

Commercial grade sodium chloride dissolved in different proportions 
in tap water constituted the eroding fluid in these experiments. The 
manufacturers of the sodium chloride supplied the data given in Table 
3.2 regarding the contents of the processed sodium chloride. The cation 
concentrations in sea salt, also included in this table, were obtained 
from Bolz and Tuve (1976). 



Table 3.2 

Cation Concentrations in Processed 
Sodium Chloride and- Standard Sea Salt 



Cation 


NaCl 


Sea Salt 


Sodium 


357460. ppm 


301720. ppm 


Calcium, Magnesium 


50. 


47770. 


Potassium 


10. 


10860. 


Phosphate 


1.0 




Iron 


0.5 





Tests were conducted for the following five salt concentrations: 
0, 1, 2, 5 and 10 ppt by weight. However, as the concentrations of the 
three most abundant cations, Na"^, Ca^"^ and Mg^'^, in the manufactured 
salt were different from those in the standard sea salt (see Table 3.2), 
the five different eroding fluids used in these experiments were not 
exactly equivalent to sea water at the various salinities. In spite of 
this, useful qualitative and quantitative information was obtained 
regarding the effect of varying dissolved salt (i.e. electrolyte) 
concentrations on the erosive characteristic of the mud. 

I 



96 

The experimental procedures used in these tests has been described 
in the previous section. Suspended sediment concentration as a function 
of time as well as the bed density profile were measured. The values of 

and Tj were 0.9 N/m^, 24 hours and 40 hours respectively (see 
Fig. 3.22). The bed shear stress during resuspension ranged from 0.14 
to 0.52 N/M'^. The bulk density and shear strength profiles for each 
salt concentration are shown in Figs. 3.13 and 3.14. 

Rates of resuspension, £, were calculated from the concentration- 
time profiles in the following manner. Smooth curves were drawn through 
the data points and values of the concentration were read off these 
curves at 0, 5, 15, 30 and 60 minutes after each change in the bed shear 
stress. Values of dC/dt were determined using a backward difference 
differentiating scheme. Values of e were calculated using Eq. 3.12. 

The logarithm of the erosion rate was plotted against the average 
normalized excess shear stress, i.e. {^^ - where is the 

average shear strength of the bed layer that was eroded by the bed shear 
stress 1^. Figure 3.28 shows these plots for the 1 ppt salinity test. 
The slope of each line, a, and the ordinate intercept, e^, were 
determined from each graph. The values of a and plotted as a 
function of depth for each salt concentration are given in Figs. 3.29 
and 3.30 respectively. 

Before evaluating these results it is appropriate to discuss 
parameters other than the salt concentration that varied from test to 
test, in order to examine the possible significance of their variance on 
the rate of erosion. The other uncontrolled parameters were the 
temperature, pH and the SAR of the eroding fluid. Also, the rotating 
annular flume does not have the facility to maintain a constant water 



97 



Z Tm =0.9N/m^ 
- Tm =24 hrs. 




0 0.05 0.1 0.15 0.2 



(^) 

Fig. 3.28. Resuspension Rate Versus Normalized Excess Shear Stress, 



98 



£ 




0 2 4 6 8 10 



. 3.29. Slope, a, Versus Depth Below Bed Surface, z, , as a 
Function of Salinity. ° 




3.30. Ordinate Intercept, e^. Versus Depth Below Bed Surface, 
Zj^, as a Function of Salinity. 



99 

temperature during the course of an experiment. As a result the 
temperature typically varied 3° to 5°C over the seven to eight hour 
duration. A temperature variation of this magnitude has been found to 
result in less than a 3% decrease in the bed shear strength (Kelly and 
Gularte, 1981) and is considered to be insignificant. Likewise, over 
the one per-cent salt concentration range used in these experiments the 
variation in pH is considered to be not significant. However, due to 
the relatively small quantities of Ca^"*" and Mg^"*" compared to that of 
Na"*", the SAR values were rather large and increased significantly with 
increasing salt concentration. For example, the SAR values varied from 
110 to 349 as the salinity increased from 1 to 10 ppt. Alizadeh (1974) 
showed that both the concentration of the electrolyte and the SAR are 
important controlling factors in the process of coagulation. 
Specifically he found that the effect of salt concentration gradually 
decreases with increasing SAR. Thus, the varying SAR values are 
considered to have had some, albeit unmeasured, effect on these 
experiments. 

Analysis of the variation in the bed density profiles with 
salinity, shown in Fig. 3.13, revealed no discernable relationship. It 
is felt that further investigations are necessary to determine if any 
relationship exists between Pg and the salt concentration of the eroding 
fluid for a stratified cohesive sediment bed. 

The bed shear strength profiles, shown in Fig. 3.14, were analyzed 
by determining the weighted depth-averaged value (weighted with respect 
to spacing, i.e. depth, between adjacent data points) of at the five 
different salt concentrations, S. The following relationship was found: 

i 



100 

-^^(S) = T^(S=0)*(S/2 + 1) for 0 £ S < 2 

(3.14) 

-^^(3) = 2t^(S=0) f or S >_ 2 

where S is in ppt. This relationship was incorporated into the bed 
formation algorithm in the following manner. The discretized value of 

at the top of the uppermost new deposit bed layer is changed 
instantaneously, i.e. during the same time-step, at every element where 
the elemental average salinity value changes. For the second bed layer 
the discretized value is not changed during the first time-step 
during which the average value of salinity changes; it is changed at the 
time-step during which the salinity changes for the second time. 
However, for this bed layer the new value is determined using Eq. 
3.14 and the second preceding value of the salinity at that element. 
This procedure is similarly repeated for the remaining new deposit bed 
layers. This method of incorporating the effect of the salinity of the 
eroding fluid on the bed shear strength profile was used in order to 
account, at least partially, for the finite amount of time it takes for 
denser (i.,e. higher salinity) eroding fluid to diffuse downward into the 
bed or for denser pore fluid to diffuse upward into the overlying 
eroding fluid. The diffusion coefficients of CI" and Na"^ in Pacific red 
clay and Lake Ontario sediment were experimentally determined to be of 
the order of 10"^ - 10"^ cm^/sec at a temperature of 24°C (Li and 
Gregory, 1974; Lerman and Weiler, 1970). These extremely small 
diffusion coefficients indicate that the rates of diffusion in 
unconsolidated sediments are generally from one half to one twentieth of 
the diffusion rates in the eroding fluid (Manheim, 1970). 



101 

For the first time-step the initial salinity value at each element 
is used to determine the values in both the unconsolidated and the 
partially consolidated bed layers, while the salinity of the pore water 
in the original settled bed layers, an input parameter in the model, is 
used to evaluate the values in this bed section. The i^ values of 
the settled bed layers are thereafter assumed to be invariant with 
respect to the salinity of the eroding fluid. The justification for 
this assumption is based on the observation that dissolved silica 
concentrations in pore waters of Lake Ontario, Erie and Superior 
sediments were, in general, invariant with respect to depth after the 
first 20 cm below the water-mud interface (Nriagu, 1978). Therefore, 
the salinity of the eroding fluid would not be expected to influence 
that of the pore fluid below the top 20 cm of the bed, which clearly 
encompasses the consolidated bed section. 

The values of a and are seen in Figs. 3.29 and 3.30, to decrease 
and increase, respectively, with increasing salinity. However, inasmuch 
as these parameters are considered to be characteristic properties of 
the sediment bed, and as the effect of salinity on another bed property, 
-c^, which is as well estimated indirectly from measured data, has 
already been incorporated into the model, it was not necessary to 
consider the variation of a and with salinity in the erosion 
algorithm. 



3.4.3. Erosion Algorithm 

A description of the redispersion and resuspension algorithms is 
given below. In both algorithms, the rate of erosion is calculated on 
an element by element basis. 



102 

A portion of the unconsolidated new deposit (UND), when present, 
will redisperse (mass erode) when t:^ is greater than the surface shear 
strength of the UNO, i.e. -^^{2^=0). The thickness of the UND that fails 
totally and is instantly redispersed is equal to z^^, where z^^ = bed 
depth at which ■^^(25) = "^5- The value of z^^^ is determined from the 
linearly varying '^f.iz^) profile in each UND layer. The value of z^^ may 
be greater than the thickness of the top layer, TLAYM(l), in which case 
more than one layer is redispersed. The dry mass of sediment that is 
redispersed, Mp, is calculated according to 



= J P(z )dz (3.15) 
0 

where Mq has units of Kg/m'^ and is considered to be the mass eroded over 
one time-step At. The contribution to the source term in the governing 
equation (Eq. 3.5) caused by redispersion is given by Eq. 3.15 divided 
by the product of the average elemental water depth and the time-step 
At. New UND layer(s) thicknesses and '^f^iz^) and p{z^) profiles are 
calculated at each time-step when redispersion occurs by subtracting z^ 
and resetting T:^iz^=0) and p(Z|^=0) equal to the respective initial 
values at z^ = Zj^^. If z^^ is calculated to be greater than the 
thickness of the entire UND, then all of this sediment is redispersed. 

For both the redispersion and resuspension algorithms, erosion is 
considered to occur only during accelerating flows, i.e. Tj^(t+At) > 
\it). Thus, even though ^{^(t+At) may be greater than x^{z^=0), no 
erosion will occur if > ^^(t+At). This stipulation for the 

occurrence of erosion, and an analogous one for deposition (as will be 
discussed in Section 3.6), is based on an interpretation of the 



103 

typically observed Eulerian time-concentration variation in an estuarial 
environment. For example. Fig. 1.3 shows a time-concentration profile 
from the Savannah River estuary (Krone, 1972). Also indicated is the 
observed correlation between accelerating flows and increasing 
suspension concentration and between decelerating flows and decreasing 
suspension concentration. Laboratory evidence (Mehta and Partheniades, 
1975; Partheniades, 1977; Mehta et^., 1982a; Parchure, 1983) suggests 
that under accelerating flows, erosion occurs without redeposition of 
the eroded sediment. Likewise, during decelerating flows, sediment 
deposits without reentrainment of the deposit. During periods of steady 
flows, erosion or deposition may occur. These two processes do not, 
however, occur simultaneously even in this case (Parchure, 1983). The 
initial condition at the inception of the steady flow period will 
determine whether erosion or deposition will occur. If the antecedent 
phase was one of acceleration, the sediment will continue to erode under 
the steady flow condition. In both cases, however, relatively short 
transient periods of simultaneous erosion and deposition sometimes do 
tend to occur (Yeh, 1979). For estuarial modeling purposes, however, 
these periods may be ignored without introducing any significant errors. 

Resuspension of partially consolidated beds (CND) occurs when: 1) 
the entire UND has been redispersed, 2) T|^(t+At) > "^^it) and 3) 'i:|^(t+At) 
> -^^(21^=0), where z^=0 is now at the fluid-CMD interface. The 
resuspension rate expression (Eq. 3.13) found by Mehta _et al_. (1982a) is 
used to determine the thickness of the CND, zu , that is resuspended 
during a time-step. At. The following iterative procedure is used to 
calculate z^^ during any given time-step. 



104 

The average erosion rate, e, for the period At is calculated as: 



I =1/2 [£(t) + e(t+At)] (3.16) 

in which 

(t+At) 

D 

e(t+At) = e^CDexpr (1)( -1)] (3.17) 

c 

where ^qH) and a(i) are the average empirical coefficients for the 
first (i.e. top) CMD layer, and 



1 2. 

'\ =^^2!:\(Zb=0) + — / ^*-^^(Zj^)dz] (3.18) 



As a first guess, z^^ is set equal to TLAY(l) = Zj^^^ (see Fig. 3.15). 

A new value of Zj^ , designated Zj^ , is calculated according to: 
* *2 




(3.19) 



where p is the average dry bed density over the first bed depth Z[^^_^. 
Then the following parameter is evaluated: 



^ - 1 = ix-il (3.20) 



A\ = . 1 = \x-l\ 

sAt 



105 

where p and e are determined using z = z . If aa. < 0.02, then Zu is 
taken to be the depth of bed eroded during this time-step. If AA. > 
0.02, then yet another new value of z^^^, designated is calculated 

using the following equation: 



Z =z + 1 (3.21) 

"*3 \, - \ 



where = P'zu /(e*At^' '^''"'^^ ^ ^""^ ^ determined using Zj, = z^ . 

^^■'"9 ^b^vo' ^'^^ entire procedure, i.e. Eqs. 3.16 through 3.21, is 
*3 

repeated until the chosen error criterion, i.e. aa < 0.02, is 
satisfied. As in the redispersion routine, new CND layer thickness(es) 
and and p(z^) profiles are determined. As before, Z|^ may be 

greater than the thickness on the top layer. Laboratory tests required 
to evaluate i^c^^b^' *^^^b^» ^""^ average values of and a for each 
CMD layer are described in Appendix D, Section D.2. 

Once the entire new deposit bed section has been eroded, the 
original settled bed, if any exists, will undergo resuspension when the 
following two conditions occur: 1) t|^(t+At) > \{X.) and 2) T[^(t+At) > 
'Cj,(Zfj=0), where Zfj=0 is now at the top of the settled bed. The surface 
erosion rate expression (Eq. 3.12) given by Ariathurai and Arulanandan 
(1978) is used to evaluate the thickness, z^,^, of the settled bed that 
is eroded during each time-step. The iterative procedure used for the 
CND is again used to solve for Zj,^, with only the expression for e being 
different,. Equation 3.16 becomes 



106 

e(t+At) = M(l)( : 1) (3.22) 

c 

where M(l) is the erodibility constant for the first layer. 

The contribution to the source term in Eq. 3.5 caused by 
resuspension is given by Eq. 3.16, with Eq. 3.17 used for the partially 
consolidated bed section and Eq. 3.22 used for the original settled bed 
section, divided by the average elemental water depth. 

In the following section, the dispersive transport of suspended 
sediments is discussed, followed by a description of the dispersion 
algorithm. 

3.5. Dispersive Transport 
3.5.1. Dispersion Mechanisms 

There have been numerous studies on the dispersion of some quantity 
(e.g. sediment) in a bounded shear flow in the thirty years since the 
work of Taylor (1953, 1954). Taylor proved that a one-dimensional 
dispersion equation can be used to represent the longitudinal dispersion 
of a quantity in turbulent pipe flow. Taylor's analysis has since been 
extended to shear flow in both rivers and estuaries. The present 
discussion is limited to dispersion in estuary flow. A brief review of 
dispersive transport theory precedes that of estuarial dispersion. 

The governing equation (Eq. 3.5) derived for the two-dimensional, 
depth-averaged movement of suspended sediment in a turbulent flow field 
includes dispersive transport terms which account for the transport of 
sediment by processes other than advective transport. Some of these 
other processes include the effects of spatial (i.e. transverse and 
vertical) velocity variations in bounded shear flows and turbulent 



107 

diffusion. Thus, the effective sediment dispersion coefficients in Eq. 
3.5 must include the effect of all processes whose scale is less than 
the grid size of the model or what has been averaged over time and/or 
space (Fischer et__al_. , 1979). For example, the effect of the actual 
vertical concentration gradient would have to be incorporated in the 
dispersion coefficients in the present depth-averaged transport model. 

Diffusion is defined as "the transport in a given direction at a 
point in the flow due to the difference between the true advection in 
that direction and the time average of the advection in that direction," 
and dispersion is defined as "the transport in a given direction due to 
the difference between the true advection in that direction and the 
spatial average of the advection in that direction" (Holley, 1969). 
Holley enunciates the fact that diffusion and dispersion are both 
actually advective transport mechanisms, and that in a given flow field, 
the relative importance of one mechanism over the other depends on the 
magnitude of the concentration gradient in the particular transport 
problem. In the governing equation (Eq. 3.5) the effective sediment 
dispersion coefficients are equal to the sum of the turbulent diffusion 
coefficients and dispersion coefficients. This approach follows the 
analysis of Aris (1956) which showed that the coefficients due to 
turbulent diffusion and shear flow (dispersion) were additive. Thus, 
the analytic expressions to be used for the effective sediment 
dispersion tensor would include, at least in some sense, both diffusion 
and dispersion. 

Fischer (1966) showed that the dispersion of a given quantity of 
tracer injected into a natural stream is divided into two separate 
phases. The first is the convective period in which the tracer mixes 



108 

vertically, laterally and longitudinally until it is completely 
distributed across the stream. The second phase is the diffusive period 
during which the lateral and possibly the vertical (depending on the 
nature of the tracer) concentration gradient is small, and the 
longitudinal concentration profile is highly skewed. The governing 
equation (Eq. 3.5) is strictly valid only in the diffusive period. The 
criterion for determining if the dispersing tracer is in the diffusive 
period is if it has been in the flow longer than the Lagrangian time 
scale and has spread over a wider distance than the Lagrangian length 
scale (Fischer et_ al_., 1979). The latter scale is a measure of the 
distance a particle travels before it forgets its initial conditions 
(i.e. position and velocity). 

Analytic expressions for the sediment (mass) diffusion coefficients 
can be obtained by analogy with the kinematic eddy viscosity. 
Specifically, the Reynolds analogy assumes that the processes of 
momentum and mass transfer are similar, and that the turbulent diffusion 
coefficient, E, and the kinematic eddy viscosity, e^, are in fact 
linearly proportional. Jobson and Sayre (1970) verified the Reynolds 
analogy for sediment particles in the Stokes range (less than about 100 
m in diameter). They found that the "portion of the turbulent mass 
transfer coefficient for sediment particles which is directly 
attributable to tangential components of turbulent velocity 
fluctuations: (a) is approximately proportional to the momentum 
transfer coefficient and the proportionality constant is less than or 
equal to 1; and (b) decreases with increasing particle size." 
Therefore, the effective sediment mass dispersion coefficients for 
cohesive sediments may be justifiably assumed to be equal to those for 
the flow itself. 



109 

Fischer ^ a]_. (1979) define four primary mechanisms of dispersion 
in estuaries: 1) gravitational circulation, 2) shear-flow dispersion, 
3) bathymetry induced dispersion and 4) wind-induced circulations. 
These four mechanisms are briefly described next. 

Gravitational or baroclinic circulation in estuaries is the flow 
induced by the density difference between the fresh water at the 
landward end of the estuary and the sea water at the ocean end. There 
are two types of gravitational circulation. Transverse gravitational 
circulation is depth-averaged flow that is predominantly seaward in the 
shallow regions of a cross-section and landward in the deeper parts. 
Figure 3.31a depicts this net depth-averaged upstream (landward) and 
downstream (seaward) transport and the resulting transverse flow from 
the deeper to the shallower parts of the cross-section. Fischer et al . 
(1979) state that "the upstream flow is expected to be concentrated in 
the deeper portions of the channel, because the upstream pressure 
gradient increases linearly with depth below the water surface." Thus, 
the interaction between the cross-sectional bathymetry and the 
baroclinic flow causes the transverse circulation. 

Vertical gravitational circulation is schematically illustrated in 
Fig. 3.31b which shows the predominantly seaward flow in the upper part 
of the flow and landward flow in the lower part. Fischer (1972) 
believes that the vertical gravitational circulation will be more 
important than transverse circulation only in highly stratified 
estuaries. 

The previously described mechanism of shear-flow dispersion is 
believed to be the dominant mechanism in long, fairly uniform sections 
of well-mixed and partially stratified estuaries (Fischer et al.. 



110 




(a) 



NET DOWNSTREAM TRANSPORT 
"NEAR THE SURFACE 



VECTICAL AOVECTION 
AND DIFFUSION 



Fig. 3.31, 




NET UPSTREAM TRANSPORT 
NEAR THE BOTTOM 



(b) 



The Internal Circulation Driven by the River Discharge in 
a Partially Stratified Estuary, (a) A Transverse Section; 
(b) A Vertical Section (after Fischer et al_. , 1979). 




Ill 

1979). Holley et aT_. (1970) applied the dispersion analysis of Taylor 
(1954) to oscillating flow in estuaries. They concluded that for wide 
estuaries, the effect of the vertical velocity distribution on shear- 
flow dispersion is dominant over that of the transverse velocity 
distribution. The exact opposite situation was found for relatively 
narrow estuaries. 

The joint influence of bathymetry and density differences on 
dispersion has already been mentioned in the discussion on baroclinic 
circulation. Other examples of bathymetry induced dispersion include: 
the intrusion of salinity or sediment in certain parts of a cross- 
section caused by the channelization of flood and ebb tides in tidal 
inlets or narrow estuaries (Fischer etal_., 1979); and the enhanced 
dispersion of a quantity (e.g. pollutant) or intrusion of salinity in 
tidal flats and side embayments, which serve as storage areas for these 
constituents, caused by the out of phase flow which occurs between the 
main channel and such features (Okubo, 1973). 

An example of a wind-induced circulation is shown in Fig. 3.32. 
The steady onshore wind causes a circulation in the wind direction in 
the shallow bay, where the less water mass per unit surface area results 
in a higher acceleration and therefore quicker response to the wind- 
induced surface stress, and in the opposite direction in the deeper 
sections of the channel. Such a circulation can cause significant 
dispersion (Fischer _et_al_. , 1979). 

For a detailed description of dispersion, the reader is referred to 
the following references: Dispersion in estuaries - Glenne and Selleck 
(1969), Pritchard (1969), Holley (1969), Holley etal_. (1970), Fischer 
(1972), Okubo (1973), Ward (1976), Fischer (1978), Murray and Siripong 



112 

(1978), Zimmerman (1978) and Fischer ^al_. (1979); Dispersion in 
channel flows - Fischer (1966), Dagan (1969), Fischer (1970), Peterson 
et al_. (1974), Taylor (1974), Ward (1974), Sumer and Fischer (1977), 
Smith (1978), Beltaos (1980a, b), Chatwin (1980) and Liu and Cheng 
(1980); Dispersion of particle matter - Sayre (1969), Jobson and Sayre 
(1970), Chen (1971), Sumer (1971) and Alonso (1981). 

3.5.2. Dispersion Algorithm 

The most important, and possibly the most difficult task in 
modeling dispersion is to determine which of the dispersion mechanisms 
are important in the estuary being modeled. For example, if the estuary 
has only a few tidal flats and shore irregularities and has a fairly 
uniform cross-section (e.g. the Delaware River), shear flow dispersion 
may be the dominant mechanism. However, if the estuary is relatively 
deep and the river discharge is large (e.g. the Mississippi River), 
gravitational circulation may be just as or even more important than 
shear flow dispersion. Unfortunately, none of the existing dispersion 
models, most of which are two-dimensional (e.g. DISPER (Leimkuhler _et_ 
al . , 1975)), can represent the combined effects of, for example, an 
irregular shoreline configuration and bathymetry, shear flow dispersion 
and baroclinic flow. 

Because of these problems in identifying, describing and modeling 
the various dispersion mechanisms which occur in estuaries, the decision 
was made to develop a dispersion algorithm for only shear flow 
dispersion that would be applicable to a wide, vertically well mixed 
estuary. Following the analysis of Hoi ley et al . (1970), it is assumed 
that the dispersion in wide estuaries is associated primarily with the 



113 

vertical shear. The limitations, which determine the applicability of 
such a dispersion algorithm, are consistent with those associated with a 
two-dimensional, depth-averaged cohesive sediment transport model. 

The dispersion tensor derived by Fischer (1978) for two- 
dimensional, depth-averaged bounded shear flow is used in the dispersion 
algorithm. The four components of this tensor are 

D^y = (UVd2/E)I^y (3.23) 
Dyj^ = (UVd2/E)Iy^ 

Dyy = (V2d2/E)lyy 

I 

in which: U and V are the root-mean-square values of u 

and V over the depth d; 
u' = u(z) - u, where u is the depth-averaged 
component of the velocity in the x- 
di recti on; 

V = v(z) - V, where v is the depth-averaged 

component of the velocity in the y- 

di recti on; 
E = mean value of the scalar turbulent 

diffusion coefficient in the vertical 

direction, E^; and 



1 „ CI C „ 

I. . = Ju. / — Ju.dCdCdC (3.24) 
IJ 0 % E' 0 ^ 



114 

in which: E = E^/E, u^- = u^-/u and C = z/d. The quantities u and v 

are the velocity deviations taken over the depth from the respective 

depth-averaged values, u and v. The values U and V represent the 
I I 

"intensity" of u and v , respectively (Fischer et_al_., 1979). The 
physical interpretation of the cross product dispersion coefficients D^^y 
and Dy^ is that a velocity gradient in the direction can produce mass 
(dispersive) transport in the Xj direction. 

Fischer (1978) notes that since in most investigations the vertical 
velocity profile, i.e. u(z) and v(z), and the vertical turbulent 
diffusion coefficient, E^, are not known with a high degree of accuracy, 
it would usually suffice to assume that the value of I^-j in Eq. 3.24 is 
a constant. The value of I^-j in various parallel shear flows ranges 
from 0.054 for turbulent pipe flow to 0.10 for laminar flow with a 
linear velocity profile over d (Fischer et_al_. , 1979). Therefore, 
Fischer recommends that a value of 0.10 be used for I^-j in Eq. 3.23. 

The following expression for E^, derived by Elder (1959) for flow 
down an infinitely wide inclined plane, is used in this analysis: 



E = Ku.zd— ) (3.25) 
^ ^ d 



where < = von Karman turbulence constant, and u^ = shear velocity. 
Therefore, E is given by 



— Id <u d 

E = - / E dz = — — = 0.067u^d 
d 0 ^ 6 ^ 



(3.26) 



115 

with K = 0.40. The values of < obtained by Gust (1976) from the slopes 
of measured clay suspension velocity profiles varied between 0.3 and 
0.4. Gust considered this variation a result of the mean flow 
experimental error of S% and not due to the presence of suspended 
cohesive sediments. Therefore he assumed < = 0.40 in his analysis. 
This is the justification for using this value of ^ in this study. 

Fischer (1966) found that in both laboratory experiments and in 
real streams that the mean value of U^/u^ was equal to 0.2. 
Substituting this value, and I^- ■ = 0.1 and Eq. 3.26 into Eq. 3.23 gives 



XX 



0.2(u)^d^ (u)^d 



D = (0.10) = 0.30- 



n.067u^d u^ 



0.2uvd^ uvd 
D = (0.10) = 0.30 



0.067u^d 



f- "f 



0.2vud ^ uvd 
-(0.10) = 0.30 



(3.27) 



0.067u^d u^ 



0.2(v)^d^ (v)^d 
D = (0.10) = 0.30 

yy 



0.067u^d u^ 



These are the coefficients used in the dispersion algorithm to model the 
shear flow dispersion of suspended cohesive sediments in a wide, well 
mixed estuary. Values of D^.j are calculated at each time step in the 
model using the specified nodal values of u, v and d. 



116 

In the following section, the depositional behavior of cohesive 
sediments is summarized, followed by a description of the deposition 
algorithm, 

3.6. Deposition 

3.6.1. Previous Investigations 

Deposition has been defined to occur when t:^ is not high enough to 
resuspend sediment material that settles onto and bonds with the bed 
surface. This process, therefore, involves two other processes, 
settling and bonding, i.e. cohesion.' Laboratory studies on the 
depositional behavior of cohesive sediment in steady turbulent flows 
have been conducted by, among others. Krone (1962), Rosillon and 
Volkenborn (1964), Partheniades (1965), Partheniades et al_. (1966), 
Migniot (1968), Lee (1974), Mehta and Partheniades (1975) and Mehta et 
al . (1982b). The results from these and other studies on the settling 
rates of cohesive sediments pertinent to the deposition algorithm 
described in Section 3.6.4 are summarized below. 

In laboratory flumes, the depositional behavior is usually 
investigated by allowing sediment suspended in a flume at a high shear 
stress to deposit by reducing the shear stress. Since the sediment 
concentration gradient in the direction of flow in usually small, the 
observed time-rate of change of the depth-averaged concentration, C, is 
due to the deposition of suspended material. The conservation of 
sediment mass can be expressed as (Einstein and Krone, 1962): 



.r (OC 
dC d s 



dt 



(3.28) 



117 

where t = time, d = flow depth, WgCO = sediment settling velocity as a 
function of C, and = probability of deposition, or the probability of 
a sediment particle or floe bonding to the bed and not being instantly 
resuspended. 

Krone (1962) conducted a series of depositional tests in a 31 m 
long and 0.90 m wide recirculating flume using mud from San Francisco 
Bay which contained approximately equal proportions of clay and silt. 
Krone postulated that P^ increases linearly with a decrease in 
according to 

P . = 1 (3.29) 

cd 

where = critical shear stress for deposition, above which no 
deposition occurs. Therefore, P^ decreases linearly from a value of 1.0 
at T^f, = 0, to 0 for > x^^. The value of was found to be equal to 
0.06 N/m^ for the Bay mud with C < 0.3 g/1. Krone found that when C < 
0.3 g/1, was independent of C. In this case, integration of Eq. 3.29 
gives 



C d s 

— = exp[ 1] (3.30) 

C„ d 

0 



where Cq is the initial suspended sediment concentration. Thus, 
according to this equation all the suspended sediment will eventually 
deposit when '^b < t j. 



118 

For 0,3 g/1 < C < 10 g/1 and for C > 10 g/1 , logarithmic laws of 
the following form were derived: 



log C = -K[log(t)] + Constant (3.31) 

where K was found to be a function of d and P^^. Krone attributed the 
variation of the depositional properties with suspension concentration 
to different forms of settling. Various forms of settling of coagulated 
cohesive sediments are discussed later in this section. 

Partheniades (1965) conducted deposition tests in a open, flow 
recirculating flume using Bay mud. He noted that for flows above a 
certain critical bed shear, the suspended sediment concentration, after 
an initial period of rapid deposition, approached a constant value, 
which he referred to as an equilibrium concentration, Cg^. The ratio 
C^„/C- = C* was found to be a constant for given flow conditions, 
regardless of the value of Cg. Whereas for bed shears even slightly 
less than this critical value, all the sediment eventually deposited. 

Partheniades et_ al_. (1966) conducted deposition experiments in a 
rotating annular flume (similar to the one at the University of Florida, 
but with mean diameter of 0.82 m and 0.19 m wide) at the Massachusetts 
Institute of Technology using a commercial grade kaolinite. Based upon 
these experiments it was concluded that Cg^ represents the amount of 
sediment that, having settled to the near bed region, cannot withstand 
the high shear stresses present there (due to insufficiently strong 
interparticle bonds) and are broken up and resuspended. In addition, 
Cgq in the fine sediment deposition tests appears not to be the result 
of an interchange between suspended and bed material as it is for 



119 

cohesionless sediment, because if such were the case, Cg^ would not be 
dependent on C^. Therefore, it follows that C^^ does not represent the 
maximum sediment carrying capacity of the flow, as it does in the case 
of cohesionless sediment, but instead may be considered to be the steady 
state concentration (Mehta and Partheniades, 1973). 

As noted by Mehta and Partheniades (1975), Krone did not observe 
Cgq in his tests because most of them were conducted at < 
wherein C would be expected to be equal to zero. It is apparent that 
the definition of ?^ must be extended to include bed shear stresses 
greater than t^^. 

Mehta and Partheniades (1975) investigated the depositional 
properties of a commercial grade kaolinite in distilled water and in 
salt water at seawater salinity (35 ppt) in a rotating annular flume 
facility at the University of Florida. Figure 3.33 shows typical 
suspended sediment concentration-time plots found in these tests. It is 
evident that a steady state concentration was reached in each test and 
that for bed shears above approximately 0.16 N/m^, the value of Cgq was 
greater than zero and in fact increased monotonical ly with increasing t:^. 

Figure 3.34 shows the ratio C*q = Cgq/Cg plotted against for all 

the tests with kaolinite in distilled water. Two important conclusions 

are obtained from this figure: 1) C*q is a constant for a given 

(and type of sediment) and is not a function of depth, d, or C^, and 

2) for "^s < . C* = 0. The first conclusion is based on the 

° "mi n ^" 

observation that the data points for all the different flow conditions 
are almost randomly scattered about a "best fit" line. The minimum bed 
shear tu > observed in Fig. 3.34 is the same as the \d value defined 
by Krone (1962), and the critical bed shear obtained by Partheniades 



120 



Cr-^ — ^ 



ojevjojco'*? 

^odcidcidcio 



X d — o> irt (Ti (o d't 



oOOOOOO 



•ddbddddo 




0) 



CD 



n3 



00 



Q 
C 



+-> 



o 

S- 

o 

4- 



0) 



(/I 
CU 

o -o 



o 

+J 

fa 



cn 



s- 




cn 



121 

(1965). As observed in this figure, found to be approximately 

'^min 

0.18 N/m^ for kaolinite in distilled water. 

In Fig. 3.35 the data of Fig. 3.34 are plotted on log-normal 
coordinates as in percent against where \ = ^b'^'^b^^-^* 

straight line through the data points gives the following relationship 
between these two dimensionless parameters: 



C =1/2 (1 + erf (— )) 
eq 

/2 



(3.32) 



with 



^a = ^'ho^ 



(3.33) 



where is the standard deviation and (v^^50 geometric mean of 

the log-normal relationship given by Eq. 3.32, and erf is the error 
function (Mehta and Partheniades, 1975). 

It was found that for all tests with kaolinite in both distilled 
and salt water, with a 50/50 mixture of kaolinite and San Francisco Bay 
mud in salt water, with only Bay mud in salt water, and for the 
reanalyzed deposition tests of Rosillon and Volkenborn (1964), 
Partheniades (1965) and Partheniades et__al_. (1968), the value of was 
0.49. Therefore, as noted from Eq. 3.33, C*^ is dependent solely on the 
value of the ratio (Vl)/{V^^50- '^^^^^ ^"^ Partheniades ( 1973) found 
for deposition tests in salt water the following relationship between 

Xin ^""^ ^Vl^SO: 



122 




123 

= 4 exp(-1.27 _ ) (3.34) 
mi n 



Mehta and Partheniades (1975) found the following dimensionless 
log-normal relationship for the variation, i.e. decrease, of the 
suspended sediment concentration v/ith time: 

^ T 

C =1/2 (1 + erf( ) (3.35) 

/2 

where 

T = logjQ(t/t5g) ^ (3.36) 

and where: C* = (Co'C)/(Co-Cgq) represents the fraction of the 
depositable sediment, C^-Cgq, deposited at any given time t, cr^ is the 
standard deviation of the log-normal relationship, and tgg is the 
geometric mean (i.e. the time at which C* = 50%). Figure 3.36 shows a 
comparison between some typical depositional data for kaolinite in 
distilled water and the log-normal relationship given by Eq. 3.35. This 
relationship was found to hold for all values of greater than 
approximately 0.25, with the exception that for very high Cg values 
(around 20-25 g/1 ) with i;^ < 1, an acceptable agreement with the 
measured data was not obtained. Good agreement was as well obtained 
between Eq. 3.35 and the data sets mentioned previously in this section. 

Taking the derivative of Eq. 3.35 with respect to time gives the 
following expression for the rate of change of C* y^ith time: 



124 




125 

dC* 0.434 exp(-T^/2) 

= (3.37) 

dt /2i t 

The standard deviation, and the geometric mean, t^g, were found 

to be functions of X]^, d, and C^. Shown in Figs. 3.37 and 3.38 are 

examples of the relationships found between these parameters. The 

following conclusions were arrived at from these and other similar plots 

given by Mehta (1973): 1) for a specific value of t^, the deposition 

rate was minimum. This tj^ value was found to vary between 1 and 2 for 

kaolinite in distilled water. The rate of deposition increased for 

values both less than and higher than this specific value, but not as 

significantly for higher values as for the lower values. However, for 

^b » deposition of suspended sediment occurred. For Bay mud 
max 

in sea water, Xu was determined to be 1.69 N/m'^. 2) for x. < 1, the 
■^max 

rate of deposition increased with an increase in d, while for Xj^ > i, 
the effect of d on the deposition rate was minimal. 

As noted, the settling velocity of suspended cohesive sediment 
particles has been found to be a function of, among other parameters, 
the suspension concentration (Krone, 1962). There appears to be at 
least three types of settling: 1) no mutual interference, 2) mutual 
interference and 3) hindered settling. For very low suspension 
concentrations, on the order of 0.1-0.7 g/1 , the aggregates or 
elementary particles settle independently without much mutual 
interference, and therefore the settling velocity is independent of C. 
For concentrations between approximately 0.3 g/1 and 10-15 g/1, the 
settling velocity increases with concentration due to the accompanying 
increase in interparticle (floe) collisions, and therefore increased 
mutual interference (Fig. 3.39). For concentrations higher than 10-15 



126 




Fig. 3.37. Log t^Q Versus for Kaolinite in Distilled Water (after 
Mehta and Partheniades, 1975). 




■k 

Fig. 3.38. Versus for Kaolinite in Distilled Water (after Mehta 
and Partheniades, 1975). 



127 




C \ I I I I I 1 I I I 

0.1 0.2 0.4 0.6 0.8 1.0 2.0 

SUSPENDED SEDIMENT CONCENTRATION, C(g/^) 



Fig. 3.39. Settling Velocity, W^, Versus Suspended Sediment 

Concentration, C, for San Francisco Bay Mud (after 
Krone, 1962). 



128 

g/1, the settling velocity actually decreases with increasing 
concentration (Figs. 3.40 and 3.41). At such high concentrations the 
sediment suspension, referred to as fluid mud or mud cake (Bellessort, 
1973), hinders the upward flux of water expelled by consolidation of the 
lower suspension (Krone, 1962). 

In the mutual interference range. Krone (1962) and Owen (1971) have 
found the following empirical relationship between the median settling 
velocity, Wg, and C: 

= Kc" • (3.38) 

where K and n are the empirical constants that depend on the sediment 
type and the turbulence intensity of the suspending fluid. Krone found 
n to be equal to 1.33 for Bay mud in laboratory experiments (see Fig. 
3.39). Teeter (1983) found n to be less than 1.0 for sediment from 
Atchafalaya Bay, Louisiana. 

Owen (1971) studied the effect of turbulence on the settling 
velocities of natural mud. No description of the sediment was reported, 
except that it was collected in the Thames River near Dagenham, 
England. A specially designed sampling instrument was used to collect 
sediment samples during both a spring tide and a neap tide. This tube 
collects undisturbed samples of suspended sediment in an estuarine 
environment, and immediately thereafter the median settling velocity of 
"natural aggregates" can be determined using the bottom withdrawal 
method described by Owen (1970). The value of n determined using this 
method was 1.1 and 2.2 for sediment collected during a spring and a neap 
tide, respectively. The turbulence intensity during a spring tide is 



129 



9 3.0 



X 

in 

£ 
E 



o 
o 
_l 

> 



I- 
I- 

u 

C/0 




SUSPENDED C0NCENTRAT10N,C(g/i) 
Fig. 3.40. Settling Velocity, W^, Versus Suspended Sediment 
Concentration, C, for Yangtze River Estuary Mud 
(after Huang et al . , 1980). 



Fig. 3.41, 



i;; 100.0 

£ 
£ 



> 
I- 



UJ 

> 



UJ 
CO 



10.0 



.0 



0. 



O.Oi- 



< 

S 0.0 i 



O.IO 




Severn estuary mud 
(saline water) 



1.0 10.0 100.0 



SUSPENDED SEDIMENT CONCENTRATION, 
C (g/^) 

Versus C for Severn Estuary Mud (after Thorn, 1981) 



130 

greater than during a neap tide. Owen therefore postulated that n was 
greater (and therefore as well) during the neap tide because the 
lower level of turbulence did not cause significant breakage of the 
aggregates; thus relatively large aggregates with higher settling 
velocities were formed. During the spring tide the higher degree of 
turbulence, and therefore greater internal shearing rates, did result in 
breakage of a significant proportion of the aggregates. Thus, small 
aggregates with lower settling rates, and therefore lower values of n, 
were formed. Owen believed that the interparticle collision rate was 
significantly high during both tides and therefore did not consider it 
very probable that the aggregate size would have been affected (i.e. 
limited) by this factor. Owen also performed standard settling tests in 
a one meter high bottom withdrawal tube using the same sediment samples 
as above, and found that varied linearly with C (i.e. n==1.0) for both 
spring and neap tide samples and that the values of were 
approximately one order of magnitude smaller than the values 
determined with the aggregate collection tube. The latter result is 
very significant in that it reveals the apparent effect of turbulence on 
the behavior (e.g. settling velocities) of sediment aggregates: larger, 
stronger aggregates with corresponding higher settling velocities are 
formed in a turbulent flow field than under quiescent conditions 
primarily because of increased collision rates due to high internal 
shearing rates. 

Migniot (1968) defined a "flocculation factor" F, given below, in 
order to quantify the effect of the aggregation intensity on Wg: 



131 

W 

F = (3.39) 

W 

where W- is the median settling velocity of the aggregates and W is 
A 

the median settling velocity of the elementary sediment particles. 
Bellessort (1973) reported that F varies with the grain size of the 
elementary (i.e. deflocculated) particles according to 



F = a^.D = (3.40) 



where D is the mean diameter of the particles in microns (10~^m), oc^ = 

250 and = 0.9, provided is measured in mm/s. Figure 3.42 shows 

this effect of the particle size on F and W. for numerous sediment 

samples at = 10 g/1 and salinity S = 30 ppt. Also plotted in this 

figure is the variation of F with D found by Dixit _eta][. (1982) using 

mud from Atchafalaya Bay, Louisiana. However, in these data Cg varied 

from 1.2 to 11 g/1 as indicated and S=0.0 ppt. Another important 

difference between the two data sets is that Bellessort measured W. 

^A 

under quiescent conditions, while Dixit _et__al_. (1982) measured this 
settling velocity under turbulent flow in the rotating annular flume. 
As observed, these data have the same slope between F and D as given by 
Bellessort, This suggests that, in general, F may be proportional to 
D albeit with different intercept values, at least for suspension 

concentrations with C- = 1.2-11 g/1 and 0<S<30 ppt. 



132 



MEAN DIAMETER (Microns) 




VELOCITY (mm/s) 



Fig. 3.42. Effect of Size and Settling Velocity of Elementary Particles 
on the Coagulation Factor of Natural Muds after Bellessort, 
1973). 



133 

3.6.2. Effect of Salinity on Deposition 

The larger, stronger aggregates of natural muds formed in a saline 
medium have been found to result in higher settling velocities (Krone, 
1962; Owen, 1970), which result in higher rates of deposition. Thus, 
the effect of salinity on the deposition of cohesive sediments may be 
quantified in terms of a relationship between salinity and the median 
settling velocity, Wg, of a particular sediment. 

Krone (1962) studied the effect of salinity and suspended sediment 
concentration on of sediment from Mare Island Strait in San Francisco 
Bay. Hydrometer analysis showed that 60% by weight of this sediment was 
in the clay size range (i.e. < 2 urn), with the remainder in the silt 
size range. X-ray diffraction and differential thermal analyses of the 
clay fraction revealed a large content of illite, montmorillonite and 
kaolinite clay groups along with small quantities of chlorite and 
quartz. The results from settling tests performed under quiescent 
conditions in one-liter cylinders showed the effect of both salinity and 
suspension concentration on (Fig. 3.43). The influence of salinity 
on is especially significant in the range 0-2 ppt, particularly for 
the 1.0 and 0.53 g/1 suspension concentrations. This result is expected 
considering the discussion presented in Section 2.2.6. One possible 
explanation for the apparent increasing influence of salinity on with 
increasing suspension concentration, as shown in this figure, is the 
following. As the suspension concentration increases, the number of 
collisions (by Brownian motion and differential settling mechanisms in 
such an experiment) would likewise increase and therefore promote the 
formation of larger aggregates with higher settling velocities. The 
lowest order aggregate that could be formed would be limited by the 

i 



134 




135 

suspension concentration, so that even with an increase in salinity (and 
therefore a corresponding increase in cohesive forces), lower order 
aggregates with typically higher settling velocities could not form due 
to the insufficient concentration of suspended particles. 

Owen (1970) studied the variation of Wj of a natural mud with 
salinity and suspension concentration. Approximately 55% of the mud was 
in the clay size range, with the remainder in the silt range. It was 
revealed that the clay fraction was composed of, in order of abundance, 
illite, kaolinite, montmoril lonite and chlorite. Settling tests were 
conducted in a two meter high bottom withdrawal settling tube. 

The results of Owen's tests are shown in Figs. 3.44 and 3.45. 
These figures show that, in general, as the salinity and suspension 
concentrations are increased, increased cohesion and interparticle 
collision result in higher coagulation rates with accompanying higher 
settling velocities. This trend corroborates that found by Krone 
(1962), except that no "leveling off" of above a certain salinity 
value was found in these tests. The decrease in above a given 
salinity and concentration, as observed in both figures, usually 
represents the onset of hindered settling. The effect of salinity on 
is seen to be diminished at suspension concentrations in the hindered 
settling range. 

Owen (1971) found a negligible effect of salinity on the settling 
velocity of natural aggregates at two different locations in the Thames 
River estuary. The salinities at the two sampling stations varied 
between 6-10 ppt and 32-26 ppt, respectively. Evidently, the effect of 
salinity on Wj at these salinities in a turbulent flow field is much 
less than that under quiescent conditions (see Fig. 3.44). This implies 



136 




SALINITY, S (PPT) 



Fig. 3.44. Effect of Salinity on Settling Velocity of Avonmouth Mud 
(after Owen, 1970). 



138 

that increased cohesion caused by the higher salinities is counter- 
balanced by the high internal shear rates which cause the aggregates to 
be broken apart. 

Deposition tests were conducted at the University of Florida in 
order to further investigate the effect on both salinity and bed shear 
stress on the settling rates of the Lake Francis mud. The salt 
concentrations utilized in these tests were: 0, 1, 2, 5, 10, 20 and 35 
ppt. The tests were conducted in the rotating annular flume with a 
water depth of 0.31 m at the following values of Tj^: 0.0, 0.015, 0.10, 
0.20 and 0.30 M/m . The initial concentration for these tests varied 
between 3,.7 and 4.7 g/1. Before the start of each test, the sediment 
and water were mixed for two hours at a shear stress of 0.90 N/m^. The 
shear stress was then reduced to the appropriate value and the 
samples of the suspended sediment were collected at 0, 1, 2, 5, 10, 15, 
20, 30 and 60 minutes after the shear stress was reduced to t^^. 
Subsequent samples were collected with lower frequency. Each experiment 
was conducted for a period of 21 hours. 

The measured suspension concentrations were plotted against time 
for each experiment. Plots of C/Cg versus time for each value at a 
salt concentration of 5 ppt are shown in Fig. 3.46. The steady state 
concentration, Cg^, for each deposition test was determined from a curve 
drawn to represent the mean variation of the concentration with time, as 
seen in this figure. The following observations were made: 1) For the 
two lowest values of x^, i.e. = 0.0, and 0.015 M/m^, the 
concentration decreased over the duration of the experiment for all salt 
concentration values, indicating that C^^ for these x^ values would 
probably have been equal to zero if the experiment had been of longer 



139 




140 

duration. 2) For = 0.05, 0.1. 0.2 and 0.3 N/m^ the concentration 
decreased rapidly during the first hour and C^q was reached more rapidly 
as the value of ■^^ increased. 3) At the four lowest values of t;|^ the 
effect of salt concentration on the deposition rate (i.e. concentration 
variation with time) was appreciable. For the two highest values the 
salt effect was much less discernable. These results seem to indicate 
that at relatively low values of cohesive forces are predominant, 
whereas at the higher values the hydrodynamic forces (i.e. disruptive 
internal shears) become at least as significant. This explanation 
follows from the results obtained by Owen (1970; 1971) and by Mehta and 
Partheniades (1975). 

The ratio Cgq/C^ was plotted against values for all salt 
concentrations (Fig. 3.47). Interpolation of the resulting plot yields 

' ^""^ believed that even though additional data 
(i.e. Cgq/Cg against i^ values) might have resulted in a different value 

of "^b . ' values would be reasonable close (probably within ± 

mi n r J 

25%). Analysis of the results from these experiments is given in the 
next section. 

3.6.3. Deposition Rates 

The product P^.W^ in Eq. 3.28 defines an effective settling 
velocity, W^, which, in general, is smaller in magnitude than since 
the range of P^ is between 0 and 1. The rate of deposition given by Eq. 
3.28 may therefore be written as 

dC s 

— = (3.41) 

dt d 



141 




Range I C 
Hindered 
Settling j 

RangelB I 
Mutual I 
Interference 



Range lA 
No Mutual 
Interference 



.Not a straigtit line in general 
because T^^ = f (C) 



Range E B 
Mutual Interference 



Range H A 
No Mutuallnterference 



0.25 1.0 



•-I 



b b bnnin 



b max 



3.48. Apparent Settling Velocity Description in Domains 
Defined by Suspended Sediment Concentration and 
Bed Shear Stress. 



142 

where is hereafter defined as the effective mean settling velocity 

for a given sediment. For the dimensionless bed shear stress less 

* * * 

than a certain characteristic value, Xj^ ^, with the range 0 ^\<'^^ 

designated as Range I, and for the concentration range C<C-^ for all 

values of (Fig. 3.48) the following empirical relationships for 

are assumed: 



Pd^,l 



for C < C 



1 



(3.42) 



s ,1 



P .KC 
d 



for < C < 



(3.43) 



( ) w (_ _ for c > C, 



250 



(3.44) 



IBv 



where W^j = median sediment settling velocity in the free settling 

range, gD^(P3/p^-l)/(18v) = and is defined by Eq. 3.29. 

Therefore, depending upon the value of C, the rate of deposition in 

Range I (see Fig. 3.48) is given by Eqs. 3.42-3.44. These three 

expressions for W^j are based upon the experimental results of Krone 

(1962), Ov^en (1971) and Bellessort (1973). Typical values for and €3 

are 0.1-0„7 g/1 and 10-15 g/1 respectively. The value of 1 was found to 

be approximately -0.6 using the settling velocities measured by Owen 

(1970), Huang etal_. (1980) and Thorn (1981). The values W. , K, n and 

s , i 

Ci can vary widely, depending upon, among other factors, the particle 
diameter, D, the type of sediment and the salinity. These parameters 
must be determined in laboratory settling tests (further discussion of 
this aspect is given in Appendix D, Section D.2). 



143 



^max ~ ^max Kin> ^""^ concentration range C 
> C^, designated as Range IIB, the rate of deposition is determined 
using a log-normal relationship (Mehta and Partheniades, 1975): 



dC -0.434 exp(-T^/2) 



dt 2y7^a^ t ^° 



* 



2.04 ^V^^ 
(l-erf( log.^C ])) {3.45) 

/2 ^«>^P(-l-27^bmin) 



where = 0.49. Eqs. 3.32-3.34 and C* = iC^-Cf iC^-C^^) have been 
substituted into Eq. 3.37. The following expression for in Range IIB 
was determined by equating Eqs. 3.41 and 3.45: 



0 



k^d exp(-T^/2) C 

= 

o^t C 



2.04 (V^^ 

(l-erf(_log^ [ ])) (3.46) 

/2 4exp(-1.27x. . ) 

bmin 

* 

where l<2 = 0.434/ (2/2Tt). For x^<l, the argument of the error function 
is set equal to zero. This expression for {and therefore Eq. 3.45 as 
well) is assumed to be valid for C>Ci for the following reason: the 
phenomenon of hindered settling was not observed in the steady-state 
deposition tests under turbulent flows performed by Mehta (1973) for 
concentrations up to about 20 g/1 . Evidently, the higher t* values that 
Mehta used in his tests prevented the occurence of this mode of 
settling, inasmuch as Krone (1962) did observe hindered settling in his 
tests which, in general, were conducted at lower values of -cb- 



144 

* 

Deposition tests with 0.25 < t:^ < 1 using San Francisco Bay mud in 

sea water and kaolinite in distilled water revealed that for suspension 

concentrations less than Cj^ ==0.1-0. 7 g/1 , the exponential law given by 

Eq. 3.41 was valid. Therefore, for C < C^^ in Range IIA, the rate of 
deposition is given by Eqs. 3.41 and 3.42 with obtained from: 



p = for 1 < 1 < x (3.47) 

d ,, b,c b b 

VJ 1 max 



s,l 



where W^^j is given by Eq. 3.46 with Cg=C=Cj^. Thus, for C<C-|^ in 

Range II is defined such that the value of ii(Cj) in Range II is 

equal to i=Pcl*^''s,l C<C-|^. Therefore, and dC/dt are continuous 

functions for all concentrations in Range II. 

Likewise, the parameter t:^ ^ is defined to be the value of -t:^ at 

which the expression for ^^(C) in Range I is equal to the same in Range 

II. Thus, Wg and therefore dC/dt are continuous functions for the 

* ★ 

entire deposition range (t;^ < 1). It is apparent that Tu is not a 

max 

constant, as it is a function of the depth-averaged concentration, C. 
* 

Solving for ^ gives 



^s II 

-^b,c = l-^- forC>C^ (3.48) 



* 



where Wgjj is given by Eq. 3.46, and W^^j by either Eq. 3.43 or 3.44 
(depending on the value of C) divided by P^. 



145 

The previously described deposition tests performed in the annular 
flume using Lake Francis sediment in water with varying salinity were 
analyzed in order to determine the combined effect of salinity and bed 
shear stress on the settling rates of this sediment. The analysis 
performed is described next. 

The settling velocity at the time when 50% of the depositable 
sediment had deposited, i.e. tsg. was determined for each experiment in 
Range I in order to quantitatively evaluate the effect of salt 
concentration, and possibly the bed shear stress, on the rate of 
deposition. This particular value of the settling velocity, designated 
as WggQ, was chosen for this analysis because it can be shown to be more 
representative of the deposition rates in the time interval of interest 
in numerical simulation (see Section 3.6.4. for discussion of this 
aspect) than either the mean or the median value. The analysis was 
performed in the following manner for the experiments in Range I (i.e. 
1^ = 0.0 and 0.15 N/m''). Equation 3.28 was integrated and rearranged to 
yield the given expression for W^jq: 



C -C 
-d ^0 ^50 

W = An ( ) (3.49) 

Pd^50 ^0%q 



where C^q = suspension concentration at time t^Q. Values of W^^q 
computed from Eq. 3.49 are shown in Fig. 3.49 superposed on Fig. 3.44, 
with lines of equal shear stress drawn as indicated. The average value 
of the initial concentration, Cq. in these experiments was 4.2 g/1 . No 
consistent trend between and W rg (e.g. increasing values with 



146 



1.0 

0.8 

_ 0.6 



1 i 111' 



T 1 — r~i — ill! 



E 
E 



>- 

o 
o 
_) 

UJ 

> 



UJ 



0.4 



0.2' 



0. 
0.08- 

Q06 
004 



0.02- 




(N/m^) 



• 0.0 

* 0.015 

Average Values 



Owen s Data 

Suspended Concentration 



o 

□ 



0.25 
1 .0 
4.0 
16.0 

32.0 



g/f 
g/-^ 



J L 



I I I 



J I I I M 



.0 20 4,0 6.0 8.0 10. 20. 

SALINITY. S (PPT) 



40. 60 80.100. 



Fig. 3.49. Effect of Salinity and Bed Shear Stress on Settling 
Velocity of Lake Francis Sediment. 



147 

increasing -^^ values for all salt concentrations) is apparent in Fig. 
3.49. This observation suggests that W^gQ may be considered to be 
invariant with respect to i^i^ in Range I. 

Due to the limited data obtained, as well as the noted invariance 
of W55Q with respect to t:^ for Range I, the values of W^gg for the two 

values were averaged for each salt concentration. These average 
values, "Ws50» ^^''^ plotted against salt concentration in Fig. 

3.49. Such an averaging procedure was performed in order to further 
investigate the effect of salt concentration on W^gg. 

A power curve relationship between Vl^gg and the salinity, S, of the 
following form was desired: 

WgggCS.C) = A W^5q(35,C) (3.50) 

where ¥55g(35,C) = KC" (K and n are defined in Section 3.6.1.) and A and 
m are empirical constants. It was proposed that when S was less than 
0.1 ppt, its value would be set equal to 0.1, so that W55g(S<0.1,C) 
would be greater than zero and in fact W55g(S<0.1,C) = Wg5g(S=0.1,C) . 
The value of V^gg at S = 35 ppt was utilized as the W55g(35,C) value. 
Least squares linear regression analysis was used to determine if the 
averaged settling velocities followed such a power curve. This analysis 
gave the following values for A and m and the coefficient of 
determination, r^: A = 0.57, m = 0.13 and r^ = 0.96. As indicated by 
this r^' value, a good agreement was obtained between Eq. 3.50 and the 
data. This confirms that, at least for these experiments and the 
analysis method employed, the effect of salt concentration on W^gg in 
Range I can be expressed by a power relationship of the form given in 
Eq. 3.50. 



148 

The function given in Eq. 3.50 is incorporated into the deposition 
rate expression for Range I and Range IIB as follows: Equation 3.50 is 
used to evaluate the settling velocity as a function of the 
concentration of dissolved salt and suspended sediment. Based on the 
variety of relationships found between W^^g and C for several of these 
experiments, the following general expressions for W^gQ in Range I and 
for C < in Range II have been incorporated into the deposition 
algorithm: 



m 

W m = A,.W .S for C < (3.51a) 

SoU 1 



I n, m^ 

^s50 " '^l*^ ^1 ^ *^ ^ ^2 (3.51b) 



I n^ m. 

^s50 " ^2 < C < (3.51c) 



2 ^ 
A-gD (- — 1) p 
250 3"^ P C n m- 

W = ( ■) ( 1) -S for C > C- (3.51d) 

pl.8 18V C3 



where K^- = A^-K^- for i=l,2, and W^,^ is the constant settling velocity for 
C < Cj^. For concentrations greater than C^^ and less than C3, where 
hindered settling begins, two different expressions of the form W^i^q = 
K'C"s'" can be used to express the variation of W^^q with C and S. Two 
relationships were revealed in this concentration range in several of 



149 

the deposition tests (e.g. see Fig. 3.50). These relationships are 
permissible in the deposition/settling algorithm. If only one 
relationship is revealed from the log-log plot of W^^q against C, C3 
must be set equal to The values of ^^ and m^ apply for C < C2 while 
A2 and m^ are for C2 ^ C < C3, and A3 and m3 apply for C > C3; if C3 = 
C2, then A2 = ^l and m2 = m^. The values of K^^ and n^ apply for < C 
< C2, while K2 and n2 are for < C < C3; if C3 = C2, then K^,^ = K2 and 
n-^ = n2. The values of K3 and n3 apply for C > C3. 

For the deposition-salinity experiments in Range II (i.e. = 

0.05, 0.1, 0.2, and 0.3 N/m ) the following analysis was performed. 

* 

Figure 3.51 shows the relationship between C^q and the salt 
concentration for the values \ > \ . where Cg > 0. Based on the 



mi n 



nature of the equal curves in this figure and taking into 

consideration the limited number of deposition tests performed at > 
* 

Tu , C is assumed to be invariant with respect to salinity. 

'-'min 



In Range IIB, the effect of salt concentration on the deposition 
rate was evaluated in the following manner. The value of dC/dt at t=t5g 
was determined for each experiment. Substituting t=t5Q into eq. 3.37 
gives the following expression for the rate of deposition at tgg: 



dC -0-^3^ Co(l<q^ 

-'50= (3.52) 

/2^ a t 
2 ^50 



The value of t5g for each experiment was determined from the log-normal 
* 

plot of C versus t, and the value of 02 was evaluated according to 
(Aitchison and Brown, 1957): 



150 



£ 
5 



Q3 



0.2 



_ 0.10 
0.08 

>: 0.06 

5 0.05 

3 004 

> 0.03- 
o 

□ 0.02 

H- 
H 

LU 
CO 



0.01 
0 



^0.0 N/m 
S -O.ppt 



T — r 

2 



"1 — r 




1 III 



02 0.4 0.6 0.8 1.0 2.0 
CONCENTRATION, Cig/i) 



4.0 



Fig. 3.50. Settling Velocity Versus Suspension Concentration for 
Deposition Test with Lake Francis Sediment. 



0.6 




=0.3 N/m 



0.2 N/rrf 



0.1 N/m^ 



0.05 N/m' 



10 15 20 25 30 

SALINITY. S { PPT) 



35 



Fig. 3.51. Variation of C^^ with Salinity and x^. 



151 




(3.53) 



* 



where tj^g and tg^ are the times at which C = 16% and 84%, respectively, 
and likewise were determined from the log-normal plot. Due to the 
limited data and observed invariance of (dC/dt)|5Q with respect to t,^, 
the values of (dC/dt)|50 were averaged for each salt concentration. The 
following functional relationship between T (given by Eq. 3.36) and the 
salt concentration, S, was obtained using a linear regression analysis: 



where b = 10.78. f = -0.33 and = 0.93. When S < 0.1 ppt, S is set 
equal to 0.1 ppt. 

The effect of salinity on the deposition rate expression for Range 
IIB was incorporated by substituting Eq. 3.54 into Eq. 3.37. The 
resulting value of the time-rate of change of concentration, dC/dt, will 
decrease monotonically at any given time with decreasing salinity, while 



at the same time, the expression for C given by Eq. 3.35 still 
approaches 0 as t-^. This methodology of incorporating the effect of 



T = Tog^QCf )BS^] 

^50 




(3.54) 



* 



salinity requires that the values of tgg and cr^, used in Eq. 3.37, be 



evaluated at a salinity of 35 ppt. 



152 

3.6.4. Deposition Algorithm 

Deposition of suspended cohesive sediment occurs when 1) the flow 

is decelerating, i.e. Xj^(t+At) < ^^^(t), and 2) when tj^(t+At) < '^b^^^- 

VJhen these two conditions are satisfied at any node, the rate of 

deposition is calculated as follows. The value of t^jj^^ is evaluated 

using Eq. 3.48. Inasmuch as the log-normal relationship was found not 

to be suitable for \ < 0.25, the minimum x^^^ value is set equal to 

0.25. Therefore, Eqs. 3.28, 3.29 and 3.51a are always used to calculate 

* * 

the value of dC/dt for 0.0 ^ \ \ c (^^"9^ ^h^^Q Eqs. 3.28, 3.47 and 
3.51a are used for Range IIA. The maximum allowable value for i^j^^^, is 
1.00; therefore, Eq. 3.45 is used to determine dC/dt in Range IIB. 

The amount of dry mass of sediment deposited per time-step element, 
Mq, is determined according to 



dC 

Mr, = ~ At d^ (3.55) 
° dt ^ 



where d_ = V2 (d (t)+d (t+At) ) and dC/dt is given by 



1 dC dC 

-[— +— ] (3.56) 

2 dt^ dt^^^^ 



in Range I and Range IIA, and 



153 

C * At 0.434 f3 
— (1 - C) [erf(Jln( ) + 1] 



in Range IIB. In Eq. 3.57. e = \[Q{t) + e(t+At)), where 9 = C, tgQ, 

Cgq and o^. Equation 3.57 was obtained by integration of Eq. 3.45 
from t=0 to t=At. The thickness of the bed formed by Mq is calculated 
in the bed formation algorithm using the procedure described in Section 
3.3.2. The sink term in the governing equation (Eq. 3.5) is given by 
Eq. 3.56 in Range I and Range IIA, and by Eq. 3.57 for Range IIB. 

For unsteady flows, as occur in estuaries, the value of C^^, which 
is the steady state value of the suspended sediment concentration found 
in laboratory tests under steady flows, is assumed to be zero. 
Nevertheless, the laboratory determined log-normal relationship for 
dC/dt, as given by Eq. 3.57, is used for Range IIB in the deposition 
algorithm for the following reasons. The time-step. At, used in the 
estuarial sediment transport problems is typically of the order 
0.1t5o<At<10t5g. Therefore, dC/dt (given by Eq. 3.57) is significantly 
greater than zero at time At. This implies that after any time interval 
At, the suspended sediment concentration, C(t+At), does not approach 
Cp_, but is assumed to be equal to the following: 



dC 

C(t+At) = C(t) At (3.58) 

dt 

where dC/dt is given by Eq. 3.57. Thus, for unsteady flow conditions, 
the rate of deposition is considered to be a function of Cgq (given by 
Eqs. 3.32-3.34). Such a consideration is required for a realistic 



154 

interpretation of laboratory deposition test results for the purpose of 
ascertaining the depositional rates in the unsteady estuarial 
environment. 

3.7. Consolidation 
3.7.1. Description 

In this section consolidation of a freshly deposited cohesive 
sediment bed is described, followed by a review of some of the research 
that has been conducted on the consolidation of sedimenting clays, and a 
discussion on the possible correlation between the density and shear 
strength of such soils. 

As described previously, an estuarial sediment bed is formed when 
deposited sediment particles and/or aggregates comprising a stationary 
suspension begin to interact and form a soil which transmits an 
effective stress by virtue of particle to particle contacts. The self- 
weight of the particles, as well as the deposition of additional 
material brings the particles closer together by the expulsion of pore 
water between the particles and/or aggregates. Thus, consolidation is 
caused by the self-weight of the sediment particles (Parker and Lee, 
1979). According to Parker and Lee, a soil is formed when the water 
content of the sediment-water suspension decreases to the fluid limit. 
Unfortunately, there is not a unique water content value for cohesive 
soils at which the suspension changes into soil (Been and Sills, 
1981). This critical water content (i.e. fluid limit) is a function of 
the initial water content of the suspension. Prior to soil formation, 
i.e. for water contents above the fluid limit, the suspension is in a 
stress-free state. 



155 

During the transition from suspension to soil, an extremely 
compressible soil framework or skeleton develops (Been and Sills, 
1981). The strains involved in this first stage of consolidation are 
relatively large. For example, Parker and Lee (1979) assumed that 
strain is one-dimensional and found that "the strain involved in the 
consolidation of an element of suspension, from say 10 g/1 (assume 
particle density 2650 kg/m-^, sea water density 1030 kg/m"^) to the 
commonly observed density of 1200 kg/ni^ would be derived from an initial 
solids fraction of 0.0038 and a final solids fraction of 0.1064 and show 
a 96% strain in the element." Lee aiid Sills (1979) state that this 
consolidation strain may continue for several days, or even months. The 
straining and upward expulsion of pore water gradually decreases as the 
soil skeleton continues to develop. Eventually this skeleton reaches a 
state of equilibrium with the normal stress of the overlying sediment 
(Parker and Lee, 1979). 

During the early stages of consolidation the self-weight of the 
soil mass near the bed surface is balanced by the seepage force induced 
by the upward flow of pore water from the underlying sediment. As a 
result, the effective stresses acting in the near surface region are 
very small and in general are not measurable (Been and Sills, 1981). As 
the soil continues to undergo self-weight consolidation and the upward 
flux of pore water lessens, the self-weight of this near surface soil 
gradually turns into an effective stress. This surface stress, and 
indeed the stress throughout the soil may first crush the aggregate 
structure of the soil and then crush the floes themselves. 

Primary consolidation is defined to end when the excessive pore 
water pressure has completely dissipated (Spangler and Handy, 1982). 



156 

Secondary consolidation, which may continue for many weeks or months, is 
the result of plastic deformation of the soil under a constant 
overburden. 

Figure 3.52 shows the variation of the mean dry bed density, p, 
with consolidation time for Avonmouth mud (Owen, 1977), commercial grade 
kaolinite in salt water (S=35 ppt) (Parchure, 1980) and for kaolinite in 
tap water (S=0 ppt) (Dixit, 1982). Noteworthy is the very rapid 
increase in p in approximately the first 48 hours, after which the 
increase was much less rapid, and the almost asymtopic approach to the 
final mean bed density, p^. Figure 3.53 shows the variation of the 
normalized mean bed density with consolidation time for the same three 
mud beds. This figure shows that, at least for these three mud beds, 
the time-variation of the degree of bed consolidation with time was 
approximately equal. 

Figure 3.54 shows the dimensionless relationship found between the 
bed bulk density, Pg, and the depth below the bed surface, z^, with 
consolidation times of the order of 48 hours for four natural muds. 
Figures 3.55a and 3.55b show the dimensionless relationship found by 
Dixit (1982) for T^^^ up to 24 hours and greater than 48 hours, 
respectively, for kaolinite beds in tap water. Figures 3.56-3.58 show 
the density profiles in Figs. 3.2-3.4 replotted on log-log paper. A 
brief review of the studies on the consolidation of clays is given 
below. 

Terzaghi (1924) developed the first theory governing one- 
dimensional primary consolidation in soils. This theory was based on 
several assumptions: 1) homogeneous soil, 2) one-dimensional 
compression, 3) one-dimensional (vertical) flow, 4) the self-weight of 



157 




TIME (hrs) 

Fig. 3.52, Variation of Mean Bed Density with Consolidation Time 
(after Dixit, 1982). 




■ 0 40 80 120 160 

TIME (hrs) 

Fig. 3.53. Variation of p/p with Consolidation Time (after 
Dixit, 1982). 



158 




3.54. z^/H Versus p/p for Avonmouth, Brisbane, Grangemouth 
and Belawan Muds (after Dixit, 1982). 



159 




Fig. 3.55. z^/H Versus p/p for Consolidation Times (a) Less Than 48 
Hours and (b) Greater Than 48 Hours (after Dixit, 1982). 



160 



X 

1 



1.0 



0.5 



0.2 













^ 4.17 






o 8.33 




o\ \ 






1 1 


^ \ 1 



0.3 0.5 



1.0 
p/p 



2.0 3.0 



Fig. 3.56. Normalized Bed Density Profiles for Thames Mud for Two 
Different Consolidation Times. 



1 

X 



1.0 



0.5 



0.2 



Ql 



005 





1 1 












<\ 












\Q 
























^\ 




Tjj(-=2.78 days 






S(ppt) 






. 0 






0 2 






X 4 






<^ 8 






' 16 






0 32 

i 


1 1 





02 



0.5 



1.0 
p/p 



2.0 



5.0 



Fig. 3.57. Normalized Bed Density Profiles for Avonmouth Mud as a 
Function of Salinity. 



161 



1.0 



0.50 



020 



0.1 



s ^ 



0.05- 



278 Days 

17.3 ppt 
16.7 g/-^ 



■0 ■■ P ^ 

(m) (m) (kq/m^) 

. 2.0 0.13 262 

o 4.5 0.32 251 

^ 70 0.58 211 

a 10.0 075 194 



0.02- 



0.0! 



0.2 



0.5 



2.0 3.0 50 



P / P 



Fig. 3.58. Normalized Bed Density Profiles for Different Bed 
Thicknesses . 



162 

the soil particles is neglected and 5) there is a constant relationship 
between void ratio and permeability and between void ratio and effective 
stress. The fifth assumption implies that the strain is assumed to be 
infinitesimal with respect to the thickness of the soil layer. All 
these assumptions limit the applicability of this theory to relatively 
stiff thin layers at large depths. Therefore, it can not be used in 
calculating the consolidation of soft sediment deposits, such as occur 
in estuaries, in which large strains are not uncommon and the 
relationship between void ratio and effective stress is not a constant 
(Einsele et_al_., 1974). 

Gibson (1967) developed the first completely general theory 

of one-dimensional primary consolidation of soils. The assumptions 
inherent in this theory are: 1) the soil is saturated and consists of 
an incompressible pore fluid and a compressible soil framework, 2) the 
pore fluid velocities are governed by Darcy's law, and 3) there is a 
unique relationship between the void ratio and soil permeability and 
between the void ratio and effective stress. The governing equation for 
this finite strain theory is 

t 

d k(e) ae 1 5 k{e) da de 1 Qe 
_ [ .] + [ ] + = 0 (3.59) 

de 1+e az s-1 az (1+e) de az S-1 at 

w 

where e = void ratio, Z = time independent material coordinate, which is 
a measure of the volume of solid particles only, k = soil permeability, 
S = soil specific weight, ^ = unit weight of the pore fluid and a' _ 

effective stress. This theory can be applied to thick soft sediment 



163 

deposits because it includes the effects of the self-weight of the 
sediment particles, is independent of the degree of strain and includes 
the constitutive relationships between e and k and between e and o 
(Cargill, 1982). 

Because of the non-linearity of Eq. 3.59, general closed form 
solutions have not yet been obtained. Therefore, numerical methods must 
be used to solve this equation for e(Z,t) for given constitutive k and 
a relationships and boundary conditions. Finite strain consolidation 
computer programs which solve Eq. 3.59 using explicit finite difference 
techniques have been developed by Gibson et__a2_. (1981), Schiffman and 
Cargill (1981) and most recently by Cargill (1982). The required 
constitutive relationships may be determined using any one of the 
following devices: geotechnical centrifuge, stress-controlled slurry 
consolidometer, pore pressure probe and nuclear densitometer (Croce, 
1982; Znidarcic, 1982). One important limitation of finite strain 
consolidation theory is that since the effective stress at the surface 
is assumed to be zero for all times, the void ratio there remains a 
constant. This differs, as mentioned previously, from the observed 
decrease in void ratio (which is directly related to p) at the 
surface of soft sediment beds with time (Lee and Sills, 1981). 

Lee and Sills (1981) assumed that the two constitutive 
relationships were linear, and that the consolidation coefficient 



dcJ 



P^(l+p) de 



(3.60) 



164 

was a constant and solved Eq. 3.59 analytically. They accounted for the 
decrease 1n the void ratio at the soil surface with time by adding an 
imaginary overburden of specified thickness on top of the actual bed. 
Comparison of measured density profiles for a clayey silt with those 
predicted by the analytical model yielded satisfactory results (Been and 
Sills, 1981). However, inadequacies of the assumptions made by Lee and 
Sills (1981) are reflected in these comparisons. Next, a discussion is 
presented on the shear strength characteristics of cohesive soils and 
the possible correlation between the density and shear strength of these 
soils. 

The shear strength of clays are due to the frictional resistance 
and interlocking between particles (physical component), and 
interparticle forces (physicochemical component) (Karcz and Shanmugam, 
1974; Parchure, 1980). Taylor (1948) states that some of the factors 
which affect the shear strength of clays are: type of clay mineral, 
water content, consolidation tine, stress history, degree of sample 
disturbance, chemical bonding (cohesion), anisotropy and exchange of 
cations. Consolidation results in increasing bed density and shear 
strength (Hanzawa and Kishida, 1981). Figure 3.59 shows the increase in 
the shear strength profile with consolidation time for flow-deposited 
kaolinite beds in tap v/ater. 

As mentioned previously, the nature and effect of consolidation on 
the shear strength profiles of cohesive sediment beds are not well known 
at present, and what limited information has been obtained is often 
contradictory (Parchure, 1980). Several researchers have attempted to 
establish a correlation between the bed density and shear strength of 
clay beds in order to implicitly evaluate the effect of consolidation on 
i^c(Z). The results of these efforts are summarized below. 



165 




Fig. 3.59. Variation of (Z|^) with for Various Consolidation 
Periods (after Dixit, 1982), 

I 



i 



166 

Figure 3.60 shows the correlation Owen (1970) found between the dry 
sediment density and shear strength for statically deposited beds of 
Avonmouth mud. Least squares analysis of the data plotted in this 
figure gave a slope of 2.44 and a coefficient of determination of r^ = 
0.83. The power expression relating and p is of the form 

\ = ^ (3.61) 

with a = 5.85x10"^ and P = 2.44. Owen considered that the correlation 
obtained between and p was satisfactory, considering the experimental 
error involved in the measurement of both these parameters. 

Thorn and Parsons (1980) likewise found a power relationship 
between and the dry sediment density at the bed surface, „ , for 
Grangemouth, Belawan and Brisbane muds in saline water. They obtained 
values of 5.42 x 10 and 2.28 for a and p, respectively. In an earlier 
study, however, these researchers observed no significant correlation 
between and p^ for Grangemouth mud (Thorn and Parsons, 1977), while 
Thorn (1981) obtained a linear relationship between i^ and p^ for mud 
from Scheldt, Belgium in 5 ppt water. 

Bain (1981) found a relationship of the following form between e 
and for Mersey and Grangemouth muds: 



^ P 



(3.62) 



167 



10.0 



X 



UJ 

m 

cr 
< 

LU 
X 

m 



5.0 



1.0 



05 



1 1 1 
c 


1 

s 


1 ! 

d 


1 1 /I ■ 


(mg/ ) 




(m) 




~ o 16.290 


32.9 


10.06 




- • 15.520 


17.0 


9.38 




▼ 17.475 


17.8 


6.98 


J3 / 

^ / 


■ 1 1 .do\J 


lb. / 


/I HA 

4.o4 


7 


9 6.705 


2.7 


9.73 


7.288 


4.6 


9.74 


/ — 


O 8.392 


8.8 


a76 ^ 




a 10.272 


16.8 


1002 ^ 


X / 


6.866 


33.3 


9.72 




A 6 810 


0 8 


9 74' 


h 














▼ 








' •Onr/ 








■ • T 








^ / 




- 




• / X 


- 






▼ ▼ p 






« 


a / 








/ oo 








/ ° 








h 






a / 










o 








ft 

X 




1 1 / 


1 


1 1 


1 1 1 



60 80 100 150 250 
DENSITY ^(kg/m^) 



350 



Fig. 3.60. Correlation of Bed Shear Strength with Bed Density 
(after Owen, 1970). 



168 

with C = 2 X 10^'^ N/m^ and ^ = -18.3 for f^ersey mud, and C = 8 x 10'' and 
^ = -6.1 for Grangemouth mud. The diversity of these relationships 
serves to emphasize that such properties of cohesive sediment soils must 
be established for each sediment studied. A description of the 
consolidation algorithm is given in the next section. 

3.7.2. Consolidation Algorithm 

Consolidation of the deposited sediment bed is accounted for by 
increasing the bed density and shear strength with time. These 
calculations are made on an element-by-element basis. The description 
given for the consolidation algorithm developed as part of this study is 
for any one given element. 

Consolidation is considered to begin after the bed formation 
process is complete, at which time the bed thickness will be maximum. 
As described in Section 3.3.2 and Appendix D, Section D.2, the 
discretized density profile at a consolidation time, T^^., of a certain 
magnitude T^^^^ (selected to be equal to two hours as measured in 
laboratory tests) is used in the bed formation routine to form the bed 
structure resulting from deposition of a given mass of sediment. The 
consolidation period for a bed begins the first time step during which 
no deposition is predicted to occur. 

After Tj^j hours of consolidation (i.e. 2 hrs.) the stationary 
suspension present on top of the bed becomes part of the partially 
consolidated bed, and therefore would undergo resuspension if subjected 
to an excess shear stress. The dry sediment mass of the stationary 
suspension is first determined, and then the bed formation routine is 
used to evaluate the thickness and structure of the partially 



169 

consolidated bed formed by this mass. This procedure accounts for the 
observation that after approximately two hours of consolidation a 
cohesive sediment bed undergoes resuspension when subjected to an excess 
shear, whereas for T^^^. < 2 hours the bed mass erodes when subjected to 
the same shear (Dixit, 1982). Should further deposition occur for T^^^ < 
2 hours, the value of T^^. is reset to zero, and the new bed formed is 
evaluated using the crushing procedure in the bed formation routine. 

The increase in bed density and shear strength with time is 
simulated to begin at T^^^, = 2 hours. This is the reason why the 
measured bed density profile at T^^ = 2 hours is used in the bed 
formation algorithm. The procedure for evaluating p{z^,t) is given 
below. 

1) First, the final mean bed density, is determined. ~„ and the 
time, Tj^^, at which p = Pa, have been shown to vary linearly with C^, 
which is defined to be the suspension concentration that would result if 
the entire partially consolidated bed was resuspended (Owen, 1970). 
is given by the following: 



pH 



(3.63) 



where p and H = mean bed density and bed thickness, respectively, of the 
partially consolidated bed at the end of the previous time step (i.e. 
t=t-At, where At = time step). The equations used for ~^ and T^ are 



170 



P = (p ) + aC 

"0 0 



(3.64) 



T . = (T , ) + bC 
dc^ dc^'^ 0 



(3.65) 



where (Poo)q and (T^^^ )q are the extrapolated (hypothetical) intercept 
values for = 0. These two parameters and a and b are empirical 
coefficients that must be determined experimentally. The value Pa, is 
defined to the value at t = T^^ at which the following criterion is 
satisfied: 



p(t) - P(t-At) 



< 10 



-3 



P(t) 



(3.66) 



with At taken to be 24 hours. 

2) The value of p(t = T^^) is determined next. The following 
relationship for p is indicated by Fig. 3.53: 

P(Tdc^/P»= 1 - f*exp(-p-T^,/T^^ ) (3.67) 

CO 

Least squares analysis of the three sets of data plotted in Fig. 3.53 
gave f = 0.845, p = 6.576 and for the coefficient of determination r^ = 
0.993. 

3) The bed density profile P(z5,T^j.) is next determined. The density 
profiles given in Figs. 3.54-3.58 can be expressed as 



171 



H-z. 
= A{ )' 



(3.68) 



where the values of A and B are functions of time for T^^ < 48 hours, 
and constants for T^^ > 48 hours. This variation of the bed density 
profile with time is depicted in Fig. 3.61. The value of z' = (H-Z[j)/H 
below which A and B are invariant with respect to time is defined to be 
^max* ^^''^ r^d^r\% that for T^^^ > 48 hours and for T^^, < 48 hours with z' 
^ ^max' P''^b^ ^H- 3.68 with A and B equal to the respective 

values for T^^, > 48 hours, while for T^^ < 48 hours and z'^^^ < z' < 1.0, 
pfZfj) is given by Eq. 3.68 with the values of A and B being functions of 
time. The values of A, B and z^jj^^ determined for the density profiles 
in Fig. 3.55a are given in Table 3.3. 



Table 3.3 

Variation of Empirical Coefficients 
the Relationship Between p{z^) and T^^^. 



Tj^( hours) 



7' 

max 



2 0.36 -1.40 0.43 

5 0.48 -0.72 0.60 

11 0.62 -0.45 0.76 

24 0.66 -0.50 0.84 

>48 0.80 -0.29 1.0 




Fig. 3.61. Variation of p(z, ) with T,^ Incorporated in Consolidation 
Algorithm. ° 



173 

Least square analysis of the data given in the above table revealed that 
all three parameters. A, B and z^j^^^^, varied with J^^ according to 



® " '^^dc'' "^dc ^ ^^^^^ ^^'^^^ 

with D = 0.32, -1.71 and 0.39, G = 0.24, -0.49 and 0.24, and r^ = 0.96, 
0.95 and 0.96 for 9 = A, B and z,|,ax» '"'^specti vely. 

Figure 3.58 shows a bed density profile for which p was measured 
below z' = 0.05. Based on the best fit line drawn through the data 
points in this figure, it is assumed for 0.0 < z' < 0.05, p(z^) is equal 
to the constant value p(Z(^=0.95H). This extrapolation of the density 
profile down to the bottom of the bed, = H, is required for two 
reasons: 1) in order to determine 9{z^=H), since the power law given 
by Eq. 3.68 cannot be evaluated at = H, and 2) in order to insure 
conservation of sediment mass in the bed. To summarize, p{z\^) is 
evaluated at a particular J^^ value as follows. For T^^ < 48 hours: 



'^"^^ n H-z, -1.71T, 

= 0.32T^^0-2^-^) dc (3^^^^ 

H 



0 24 "-^b -1.71(48)-1-^^ 
= 0.32(48)"^^ -) (3.71) 



H 



174 

for 0.05 < z < z^^', 

max 



^^^b^ 0.24 -1.71(48)"°*^^ 



= 0.32(48) (0.05) (3.72) 



= 1.75 for 0.0 < z < 0.05 
0.24 

where z^^^ = 0.39 T^^ . For T^^^. > 48 hours, Eqs. 3.71 and 3.72 are 
used with z'^^^ = 1.0. The discretized bed density profile used in the 
layered bed model is changed using Eqs. 3.70-3.72 to reflect the 
increase in bed density due to consolidation. 

4) The thickness of the bed, H, is reduced to account for the expulsion 
of pore water during consolidation, and to insure that the mass of 
sediment in the bed is conserved. The rate of change of H with time is 
given by 



dH H dp 

dt=--dt (■^•73) 

P 



Using first order finite difference approximations for the two 
derivatives, Eq. 3.73 becomes 



p(t-At) - p(t) 

H(t) = H(t-At) {l+2( )} (3.74) 

p(t) + p(t-At) 



175 

where H(t) = bed thickness at the current time step and H(t-At) = bed 
thickness at the previous time step. The thickness of each bed layer is 
adjusted as follows: 

H{t)-H(t-At) 

T.(t) = T.(t-At){H } (3.75) 

^ ^ H(t-At) 

where T^- = thickness of the i-l!l bed layer. 

5) If further deposition occurs when T^^ > 2 hours, a new sediment bed 
is formed on top of the existing partially consolidated bed. To 
simulate the occurrence of such repeated periods of deposition, as 
typically occurs in estuaries due to the oscillating tidal flow, the new 
deposit portion of the bed model is further divided into a finite number 
of strata (Fig. 3.62). The top stratum may be composed of a stationary 
suspension and partially consolidated bed, whereas the buried strata are 
composed of just partially consolidated beds. The degree of 
consolidation of a particular stratum (in relation to that of the other 
strata) is accounted for by using a separate T^^ for each stratum. The 
bed density profile for the 1— stratum as a function of T^^.., where the 
sub-subscript i refers to the i~ stratum, is determined as follows. 
Step 1 is performed with H = total bed thickness of all strata and 7 = 
mean bed density of all strata. Step 2 is repeated for each stratum. 
Thus a separate value of p is determined for each stratum. Steps 3 and 
4 are likewise repeated for all the strata, with the total bed thickness 
used for H and a separate value of p used for each stratum. 



176 



Strata 



X 



® 



iz 



X 



dc 



nr 



ir 



'dc 



IE 



I © 
Tdci 



Bed Surface 



Settled Bed 



z = I.O 

UNO 

PCND ^ 



PCND 



PCND 



PCND 



•z =0.0 



"^dc "^dc ^ "^dc Ti-ip 

ULj ULjj UCjjj UCj2 



Fig. 3.52. Bed Schemati zation Used in Bed Formation-Consolidation 
Algorithms . 



177 

Due to the extremely limited number of studies on the nature of 
shear strength profiles in cohesive sediment beds, the variation of the 
bed shear strength profile "^^^^b^ ^^'^'^ "'"dc ^'^ determined indirectly by 
use of a functional relationship between p and x^. The relationship 
found between these two parameters (Eq. 3.61) by Owen (1970) and Thorn 
and Parsons (1980) is used in the consolidation algorithm to account for 
the increase in with increasing T^^^. 

The empirical coefficients used in the consolidation algorithm 
(i.e. a, p, (p„)g, (T(J(;^)q, a, b, f, p. A, B, and z'^^^) must be 
determined by performing laboratory consolidation tests. A 
brief description of a test procedure is given in Appendix D, Section 
D.2. 



CHAPTER IV 
MODEL DEVELOPMENT 



4.1. Introductory Note 

This chapter begins with a review and evaluation of previous 
cohesive sediment transport models, then follows with descriptions of 
the cohesive sediment transport model developed during this study, the 
finite element formulation used in the model, and lastly, the 
convergence and stability characteristics of the model. 

4.2. Review of Previous Models 

One of the first cohesive sediment transport models was developed 
by Odd and Owen (1972). This was a two-layered, one-dimensional coupled 
model which simulated both the tidal flow and mud transport in a well- 
mixed estuary. The two layers were of unequal thickness, with uniform 
properties (e.g. flow velocity, suspension concentration) assumed for 
each layer. A rectangular flow cross-section was also assumed. The 
equations of motion and continuity for each layer were solved using a 
finite-difference formulation, while the advection-diffusion equations 
governing the transport of suspended sediments in two layers were solved 
using the method of characteristics. Erosion and deposition of sediment 
were simulated in this model. 

O'Connor and Zein (1974) developed an un-coupled two-dimensional, 
laterally averaged suspended sediment model which solves the advection- 

178 



179 

diffusion equation using an implicit finite-difference method. 
Horizontal eddy diffusion is neglected, as is the assumed negligible 
vertical water motion in comparison with the sediment settling 
velocity. The settling and erosion of sediment was accounted for in 
some of the described model applications by modifying the vertical 
sediment diffusion coefficient. The model is strictly applicable to 
quasi-steady depth and flow conditions. It was later modified to 
include unsteady (tidal) flow conditions by O'Connor (1975). 

Ariathurai (1974) and Ariathurai and Krone (1976) developed an un- 
coupled two-dimensional, depth-averaged sediment transport model which 
uses the finite element method to solve the advection-dispersion 
equation. This model simulates the erosion, transport, aggregation and 
deposition of suspended cohesive sediments. Aggregation is accounted 
for by determining the sediment settling velocity as a function of the 
suspension concentration. Required data include the two-dimensional, 
depth-averaged velocity field, dispersion coefficients, and the sediment 
settling and erosion properties. Ariathurai et al_. (1977) modififed 
this model to solve the two-dimensional, laterally averaged suspended 
sediment transport problem. This latter model was verified using field 
observations in the Savannah River Estuary. 

Kuo^al_. (1978) developed a two-dimensional, laterally-averaged, 
coupled model which simulates the motion of water and suspended sediment 
in the turbidity maximum of an estuary. The vertical dimension is 
divided into a number of layers, and a finite difference method is used 
to solve the equations of motion, continuity and sediment mass balance 
for each layer. Erosion and deposition are accounted for in the mass 
balance equation for the bottom layer. 



180 

Koutitas and O'Connor (1980) developed a three-dimensional 
suspended sediment transport model which solves the advection-diffusion 
equation using a mixed finite difference-finite element method. Central 
finite differences are used in the horizontal directions while linear 
finite elements are used in the vertical direction. The two horizontal 
turbulent diffusion coefficients and the vertical water velocity are 
assumed negligible in comparison with vertical eddy diffusivity and 
sediment settling velocity, respectively. The source/sink term which 
accounts for the erosion and deposition of sediment is not included in 
the governing equation. 

Cole and Miles (1983) describe a two-dimensional, depth-averaged 
model of mud transport which solves the advection-dispersion transport 
equation by a finite difference method. Deposition and dispersive 
transport are simulated, but erosion is not. 

None of these fine sediment transport models consider the following 
two factors. 1) Consolidation of the mud bed and the effect this has on 
the erodibility when the bed is subjected to an excess bed shear 
stress. 2) The effect of salinity variation (e.g. in the mixing zone 
between fresh and sea water in estuaries) on the processes of erosion 
and deposition of cohesive sediments in a turbulent flow field, since 
the empirical laws used to determine the rates of erosion and deposition 
were derived using empirical evidence from laboratory experiments 
conducted in natural or artificial sea water. In addition, the 
empirical laws of erosion and deposition used in these models cannot be 
considered to be "the state-of-the-art" even for sea water, as a 
considerable number of laboratory tests conducted since these laws were 
proposed have revealed new evidence on the erosional and depositional 



181 

behavior of cohesive sediments. For example, the empirical erosion rate 
expression used in the existing models is given by Eq. 3.13, which, as 
described in Chapter III, has been found to be applicable only to 
settled beds and not to partially consolidated beds. Likewise, the 
empirical deposition rate expression used in these models (Eq. 3.50) is 
limited to only a small percentage (e.g. approximately 20% for kaolinite 
in tap water) of the bed shear stress range over which subsequent 
laboratory tests have shown that cohesive sediments deposit under steady 
flow conditions (Mehta, 1973). 

4.3. Model Description 

The cohesive sediment transport model, referred to hereafter as 
CSTH-H, developed during this study is a time varying, two-dimensional, 
un-coupled finite element model that is capable of predicting the 
horizontal and temporal variations in the depth-averaged suspended 
concentrations of cohesive sediments and bed surface elevations in an 
estuary, coastal waterway or river. In addition, it can be used to 
predict the steady-state or unsteady transport of any conservative 
substance or non-conservative constituent, if the reaction rates are 
known. CSTM-H simulates the advective and dispersive transport of 
suspended or dissolved constituents, the aggregation, deposition and 
erosion of cohesive sediments to and from the bed, respectively, and the 
consolidation of the bed. 

CSTM-H is composed of the algorithms and layered bed model 
developed in Chapter III integrated into a modi_fied version of the 
finite element solution routine developed by Ariathurai (1974). A 
description of this finite element formulation and the modifications 



182 

made to it is given in the next section. A synopsis of the operations 
performed by CSTM-H during each time step is given below. The flow 
chart given in Appendix C, Section C.3 depicts the step-by-step solution 
procedure incorporated in CSTM-H. 

The average bed shear stress induced by the turbulent flow velocity 
of the suspending fluid is calculated for each element. Then the amount 
of sediment, if any, that was deposited onto or resuspended from the bed 
in each element during the previous time step is determined using the 
deposition and erosion algorithms, respectively. The dispersion 
algorithm then calculates the values of the four components of the two- 
dimensional sediment dispersivity tensor. Using these values and the 
prescribed velocity field and concentration boundary conditions, Eq. 3.5 
is solved for the suspended sediment concentration at each node for the 
next time step. The new bed elevation in each element is determined by 
adding or subtracting the thickness of sediment deposited onto or 
resuspended from, respectively, the bed profile that existed during the 
previous time step. Lastly, the consolidation algorithm calculates for 
each element the increase in bed density and shear strength and the 
decrease in bed thickness due to consolidation during the previous time 
step. 

The following five types of data are required to apply CSTM-H to a 
particular water body: 1) input/output and transient control 
parameters, 2) finite element grid of the system to be modeled, 3) two- 
dimensional depth-averaged velocity and salinity fields, 4) 
concentration initial and boundary conditions and 5) properties of the 
cohesive sediments in the water body to be modeled which characterize 
their erosion, deposition and consolidation. The user's manual in 



183 

Appendix C, Section C.4 lists the required information in each of these 
five data groups. A brief description of the five groups is given 
below. 

The required input/output and transient control parameters include 
the time step size, the degree of implicitness used in solving the 
temporal problem, and code arrays which specify at which time steps new 
values of various parameters (e.g. depth of flow) are read in. These 
transient code arrays also specify the type of output (e.g. nodal 
concentrations and/or discretized bed profiles), if any, required at 
each time step. 

The finite element grid of the water body to be modeled is defined 
by the number of elements and nodes the water body is divided into, the 
two horizontal coordinates of each node point, and the number of nodes 
which form each element. Quadratic and/or triangular elements with 
curved sides may be used in CSTM-H. A finite element grid generating 
program is a helpful tool in generating and modifying a finite element 
grid. This is especially true when a relatively large body of water 
(e.g. Tampa Bay) is being modeled. The advantages of using the finite 
element method over the more conventional finite difference method in 
estuaries and other similar bodies of water are delineated in the next 
section. 

The velocity field is defined by the two horizontal components of 
the depth-averaged flow velocity and the depth of flow at each node and 
time step. The only practical methods available today to determine the 
velocity field in the detail required by an un-coupled transport model 
such as CSTM-H to model an estuary or other prototype water body are 
physical and mathematical models. The advantages and disadvantages of 



184 

each are well documented, and therefore will not be discussed here. 
Numerous two-dimensional hydrodynamic mathematical models have been 
developed; these are likewise well documented in engineering 
literature. The FESWMS hydrodynamic model developed by the Water 
Resources Division of the U.S. Geological Survey is particularly well 
suited for modeling the velocity and salinity fields because this model 
uses the same basic finite element formulation (i.e. isoparametric 
quadrilateral and/or triangular elements with parabolic sides) that is 
used in CSTM-H. Therefore, the same grid can be used in both models. 
The salinity (i.e. density) field needs to be determined for water 
bodies where spatial and/or temporal variations in the salinity occur, 
in order to model the effects that variations in the salinity have on 
the erosion and deposition characteristics of cohesive sediments. A 
description of the method used in CSTM-H to evaluate the bed shear 
stress over a cohesive sediment bed using the two depth-averaged 
horizontal velocity components is given later in this section. 

The depth-averaged suspended sediment concentration must be 
specified at each node at the start of the modeling effort (initial 
conditions). Boundary conditions (i.e. depth-averaged suspension 
concentrations or normal concentration flux) are required for all 
external v/ater boundaries of the system being modeled. For the nodes 
which define such external water boundaries at which no concentration 
boundary conditions are given, CSTM-H assumes that the spatial 
concentration gradient is zero. The boundary conditions at the free 
water surface and the bottom are expressed by Eqs. A. 35 and A. 36, 
respectively. Equation A. 35 expresses that there can be no net rate of 
sediment transport across the free water surface. The bottom boundary 



185 

condition (Eq. A. 36) expresses that eroded sediment material is 
transported in the vertical direction away from the bed by turbulent 
diffusion and that deposited sediment becomes part of the bed. 
Equations A. 35 and A. 36 are actually included in the governing equation 
(Eq. 3.5) since it is vertically integrated from the bottom to the free 
water surface (see Appendix A). 

The sediment parameters which prescribe the erosion, deposition, 
bed formation and consolidation characteristics of the sediment in the 
water body to be modeled are described in Chapter III, Sections 3.3, 
3.4, 3.6 and 3.7 and in Appendix C, Section C.4. The field data 
collection and laboratory sediment testing programs recommended for 
obtaining the data required by CSTM-H are described in Appendix D. 
Descriptions of some additional functions incorporated in CSTM-H are 
given below. 

The settling velocity in Range IC is a function of the kinematic 
viscosity, v (see Eq. 3.59d). The following equation for v as a 
function of the mean water temperature, (which is read in the fifth data 
set above) was determined with data obtained from Bolz and Tuve (1976), 
using least squares linear regression analysis: 

V = 1.7017x10"^ • exp(-0.0251T ) (4.1) 

w 

where T^ is the mean water temperature in degrees Celsius and v has 

units of m /s. The coefficient of determination for this equation is 
2 

r =0.994, which indicates a good agreement between Eq. 4.1 and the data. 

The density of the suspending fluid (in Kg/rn^) is calculated at 
each node as a function of T^ and nodal salinity value using the 
following empirical equation (Wilson and Bradley, 1968): 



186 

P = 1000. 0*(0. 702 + 100. 0*(17. 5273 + O.llOlT - 
w w 

0.000639T^ - 0.039986S -0.000107T S) 
w w 

(5881.913 + 37.592T - 0.34395T ^ + 

w w 

2.2524S)"^) (4.2) 



where S = salinity in ppt. Incorporating, in addition, the effect of 
the suspended sediment on the local water density gives (MacArthur, 
1979) 



e,w w s w 

'^s 



where Pg ^ ~ effective local water density = f(T^,S,C), and p^^ is given 
by Eq. 4.2. At each time step where nodal salinity values change, new 
nodal values of p^ and Pg ^ are calculated. At each time step where 
nodal concentrations change, new nodal values of pg ^ are determined. 
The values of pg ^ are used in CSTM-H to calculate the nodal values of 
the bed shear stress tj^, while the values of p^ are used in determining 
the nodal values of the dry density, p, which are used in the erosion, 
bed formation and consolidation algorithms. 

The bed shear stress is calculated at every node using the 
following relationship between and u^ = friction velocity: 



\ = Pe,w"f^ 



(4.4) 



187 

Modal values for are determined as follows. The magnitude of the 
depth-averaged velocity vector, U, at each nodal point is given by 

- 2 2^/2 

U = (u^ + v^) (4.5) 

where u and v are the two depth-averaged velocity components given as 
input data. The vertical velocity profile for a fully developed two- 
dimensional, bounded shear flow over a cohesive sediment bed, which 
Mehta (1973) and Gust (1976) have found to be hydrodynamically smooth, 
is given by (Chri stensen, 1977) 



U(2) •^f^ 

= 5.5 + 2.5 In ( 5.29) 



(4.6) 



where U(z) is the horizontal velocity component at an elevation z above 
the bottom. Integrating Eq. 4.5 over the local depth of flow, d, gives 



U 

— = 2.5 In (- 



3.32 u^d 



- 17.56) 



(4.7) 



Using the value of U determined by Eq. 4.5, a Newton-Raphson iteration 
scheme is used to iterate for the value of u^. Then tj^ is determined 
using Eq. 4.4. 

A description of the FORTRAN computer program of CSTM-H is given in 
Appendix C. In Sections C.l and C.2, the functions of the main program 



188 

and subroutines are respectively described. A flow chart and user's 
manual are given in Sections C.3 and C.4, respectively. In the next 
section, the finite element routine used is described. 

4.4. Finite Element Formulation 
4.4.1. Introductory Mote 

The finite element method has been used to solve the governing 
equation (Eq. 3.5). This method is a numerical analysis technique for 
obtaining approximate solutions of differential equations. The 
discretization procedures used reduce the equation to be solved to one 
with a finite number of dependent variables by dividing the continuous 
solution domain into a number of elements and by expressing the 
dependent variable in terms of approximating interpolation (i.e. shape) 
functions within each element. The values of the dependent variable at 
node points are used to define the interpolation functions. Node points 
are usually located on the boundaries of elements and are used to define 
the connection between adjacent elements. The number and location of 
the node points must be chosen such that continuity of the dependent 
variable across common boundaries of adjacent elements is achieved 
(Zienkiewicz, 1977). The behavior of the dependent variable within each 
element is defined by the values of the dependent variable at the nodes 
and the shape function. Then the error which results from the use of 
the approximate dependent variable at each node in Eq. 3.5 is 
minimized,, This procedure results in a set of simultaneous equations 
which are solved for the unknown nodal dependent variables at the next 
time step,, A detailed description of the method is presented by 
Zienkiewicz (1977). 



189 

This method is preferred over the finite difference method because 
derivative boundary conditions do not require special treatment in the 
finite element method as they do in the former. It is a particularly 
advantageous method to use in estuarial transport problems because of 
the ability to use arbitrarily shaped elements. 

Quadrilateral and/or triangular elements may be used in CSTM-H in 
which a quadratic function approximation is used to describe both the 
intra-element spatial variation of the geometry and suspended sediment 
concentration. Therefore, the elements are isoparametric and may have 
curved sides. 

4.4.2. Shape Functions 

The global and local element coordinate systems are shown in Figs. 
4.1a and 4.1b, respectively. The global x-y coordinate system is 
continuous over the entire solution domain, while the local element ?,ri 
coordinate system applies only within an element. The local coordinate 
systems for a quadratic quadrilateral and quadratic right triangular 
element are shown in Fig. 4.1b. The local or area coordinates for the 
corner nodes 1, 3 and 5 of the triangular element are (0,0), (1,0) and 
(0,1), while those for the corner nodes 1, 3, 5 and 7 of the 
quadrilateral element are (-1,-1), (1,-1), (1,1) and (-1,1). 

Because three nodes are used along each edge of the triangular and 
quadrilateral elements, quadratic shape functions are required. The 
quadratic shape functions in CSTM-H determine the values of both the 
dependent variable, C, and the element geometry. Thus, the elements are 
isoparametric (Zienkiewicz, 1977). There is one shape function, N^-, for 
every node in a given element. Thus, for triangular elements there are 



190 



7 6 5 




I 2 3 




(a) Global Coordinates 




5(0.1) 




(b) Local Coordinates 



Fig. 4.1, Global and Local Coordinates 



191 

six shape functions while for quadrilateral elements there are eight. 
The shape functions are functions of the local coordinates C and ti and 
the values of C and ri at the nodal points. The quadratic shape 
functions for quadrilateral and triangular elements are given in Table 
4.1. The parameters C^- and ti^. in this table are the nodal 
coordinates. For example, for a quadrilateral element, r\. = -i, -i 
for node 1, while for a triangular element, 5^-, t\. = 1, 0 for node 3. 

The dependent variable, C, is approximated as the following 
function of the unknown nodal point concentrations, C^- , and the shape 
functions,, N-: 



i=n 

C. = 2 N.C. 
J i=l ' ' 



(4.8) 



where Cj = approximate suspended sediment concentration at any location 
inside the jth element, and n = number of nodes forming the jth element. 

Likewise, the global coordinates x and y are approximated as the 
following functions of the global nodal point coordinates, x^- and y^-, 
and the shape functions, N-: 



i=n 

X = 2 N.X. 
i = l ^ ^ 



(4.9) 



192 



Table 4.1 
Quadratic Shape Functions 



Quadrilateral Element 
Shape Function Node Number 



]. = (l+^C.)(l+TlTl.)(^|.+TlTl.-l)/4 

Corner Nodes 1,3,5,7 



M^. = (l-^^)(l+TiTi^. )/2 Midsection Nodes 2,6 



N^. = (1-Ti^)(l+?C. )/2 Midsection Nodes 4,8 



Triangular Element 
Shape Function Node Number 



N. = 4^.1,, (1-Ti^. )(l-€^.-Ti^. ) Midsection Node 2 



N^. = 4 l.r]./{l.+r].) Midsection Node 4 



= 4 r]^{l-l.-r].){l-l.) Midsection Node 6 



N. = U-l^--n.){l-2{l.+r\.-2l.r\.)} Corner Node 1 



N. = -2lA/2-2r]^-l.+r].^+^.l.+r]./{l.+r].)} Corner Node 3 



2 

^1 = ^T^l. -Ti . +1. +C. Ti. +C. / ( 5. +n. ) } Corner Node 5 



193 

The shape functions are used for two additional purposes: 1) to 
transform from the global coordinate system to the local element 
coordinate system and 2) to transform the derivatives of C with respect 
to X and y to the local element coordinates. To perform these 
transformations, the derivatives of with respect to x and y are 
needed. These are derived below. 

The derivatives of N^- with respect to x and y are given in terms of 
the derivatives of N^- with respect to the local coordinates C and t) 
using the chain rule of partial differentiation (Zienkiewicz, 1977) 



aN. ay aN. ay aw. 

ax ari a^ a^ a-n 



= [— • • — 1 ' iJf^ (4.10) 



aN^. ax aN^. ax m. 

— = C— • • — } ' iJi'^ (4.11) 

dy aC dri at) a^ 



where |Jl is the Jacobian given by 



1J| = 



ax 


ay 


a^ 


ac 


ax 


A 

ay 




ari 



(4.12) 



The derivatives of N^. with respect to I and ri for both quadrilateral and 
triangular elements are given in Table 4.2. 



194 



Table 4.2 
Derivative of Shape Functions 



Derivative 



Quadrilateral Element 



Node Number 



^^.(l+TlTi^.)(2^S^.+r)ri^.)/4 



Corner Nodes 1,3,5,7 



Midsection Nodes 2,6 



^i,.(l-ri^)/2 

1 



Midsection Nodes 4,8 



n^.(l+CC.)(2TiTi.+^^.)/4 



Corner Nodes 1,3,5,7 



^. (l-r)/2 



Midsection Nodes 2,6 



Derivative 



Midsection Nodes 4,8 



Triangular Element 
(6 = Kronecker Delta) 



Node Number 



4[l-2(C+Ti.)+2^.Ti+n.^]6.^. 

4Tl.[lH./(C.+ri.)]/(^.+n.)6.^. 

-4Ti.(2-2C.-..)6.. 

( -3+45 . +8T1 . -81. ri. -471^ ) 6. . 

{N„ / ^ -2C . [ - 1+Ti . +n / ( 5, +ri . ) 2 ] } 6 

-2ri. [-2+21 .+T1 .+1/ ( 6 .+n . ) +C. / ( 5.+ti. ) ^1 6. . 

-45.(2-5.-2..)6.. 

4?^.[l+n./(C.+n.)]/(5.+ri.)6.^. 

4(l-?.)(l-5.-2Ti.)6.^. 

{-3+4Ti.+8C.-8C.ri.-4C^?)6_ 

-2C^ [-2+2T1 . +C . +1 / ( I, +n . ) +n . / { I. +T1 . ) ^ ] 6 . ^. 

{N5/T15-2T1. [-1+5. +C. / ( 5. +T1. ) 2] I. 



2 
4 

6 
1 
3 
5 
2 
4 
6 
1 
3 
5 



195 

The four components of the Jacobian given by Eq. 4.12 are equal to 



— = 2 X. — = Z X. 

i=i ac ^ an i=i an ^ 



(4.13) 



ay aM 5y . aN . 

i=n 1 i=n 1 

ac 1=1 ac an i=i an 



Likewise, the derivatives of C with respect to C and n are equal to 



ac . „ aw. ac . an. 

i=n 1 i=n 1 

— = 2 C. — = Z c, (4.14) 

ac i=i ac ^ an i=i an ^ 



4.4.3. The Galerkin Weighted Residual Method 

The Galerkin weighted residual method has been used to solve the 
governing equation. This method requires that the summation of weighted 
residuals over the entire solution domain be equal to zero when the 
shape functions are used for the nodal weighting factors. The residual, 
r, results from applying the governing equation to the element subdomain 
using the approximate suspension concentration C instead of the actual 
concentration C. In order for C to satisfy all the stipulated boundary 
conditions, the sum of the normal concentration fluxes from adjacent 
elements and any source or sink must be equal to zero on all internal 
and external boundaries in the solution domain. This condition may be 
expressed mathematically as (Ariathurai and Krone, 1976): 



q^- + q^- + q^- 



196 

= 0 i=l....,NL 



(4.15) 



where 

q| = outward normal flux from one element 

= inward normal flux from adjacent element 
qf = normal flux from source/sink on the ith boundary 
NL = number of element interfaces and external boundaries 

The formulation of the Galerkin method can be expressed mathematically 
as 



j=NE k=NL 

2 / H.rdA + 2 / N.RdC = 0 (4.16) 
j=l A k=l C ^ 



where Ag = element subdomain, R = residual which results from the use of 
C in Eq. 4.15 and 5 = variable length along the k^ boundary. 

The governing equation used by Ariathurai (1974) was Eq. 3.5 with 
the off-diagonal dispersion coefficients, and D„„, equal to zero. 
This formulation was modified to include the terms involving D^^y and 
Dyj^. The following development is for the modified formulation 
incorporated in CSTM-H. 

Substituting the expressions for r and R into Eq. 4.16 yields 



j=NE ac ac ac a ac 

2 //. N . { — + u — + v (D — + 

j=i '^a at ax ay ax ^^ax 



197 



ac a ac ac 
D — ) (D — + D — ) - S } dxdy + 

^ydy dy >^ydx yyay 



k=NL 

2 (q^ + q- + q^) dC = 0 



k=l 



(4.17) 



The term Q = _L + S in this equation is taken to be an instantaneous 
at 

constant. This approach, which transforms Eq. 3.5 into an elliptic 
equation, results in a more efficient computational scheme. 
Using Pick's law, the fluxes in Eq. 4.15 are given by 



dc ac ac ac 

(D — + D — ) n + (D — + D — ) n 
XX 3^ xy^y X yx^^ yy^^ y 



(4.18) 



where n^, ny = x- and y- components of the outward normal to the 
boundary surface of Ag. 

The second derivative terms in Eq. 4.17 require continuity of the 
first derivatives to insure convergence. This would involve solving for 
the unknown first derivatives of C. To avoid this added complication, 
the second derivatives are transformed to first derivatives using the 
divergence theorem. The first term in Eq. 4.16 may be expressed in 
vector form for one specific element as 



n (v'vc - v(Dvc) + Q) dv = 0 



(4.19) 



198 

where -V = solution domain, D = sediment dispersivity tensor, and V = 
two-dimensional vector operator. The dispersive flux vector F is equal 
to DVC. Thus, the second term in the integrand of Eq. 4.19 becomes 



NV(DVC) = NVF = V(NF) - (VN) * F (4.20) 



The divergence theorem states that 

V • (NF) dV = NF • ndS (4.21) 

where n = outward normal to the surface S of the domain -V. Substituting 
Eqs. 4.20 and 4.21 into Eq. 4.19 gives 

VN 

L_N (V • VC + — • (DVC) + 0) dV - / N DVC • ndS (4.22) 

^ N ^ 

Substituting Eqs. 4.18 and 4.22 into Eq. 4.17 yields for locally 
constant dispersion coefficients 



A. A 



j=NE ac ac ^1 ac 

2 //, [N (Q + u— + V—) + (D^ — + 

j=i e ax ay ax ^^ax 



ac K K 



2 N. qj dC = 0 (4.23) 
I<=1 ^ " 



19? 

This equation may be expressed for a singular element by the 
element matrix differential equation 



a{c} 

[k] {C} + [t] + {f} + Cb] {O = 0 (4.24) 

at 



where 



[k] = element steady-state coefficient matrix 
[t] = temporal matrix 

{C} = vector of unknown nodal concentrations 

{f} = element source/sink vector 

[b] = boundary or element load matrix. 

Equation 4.24 is evaluated for each element with the element load 
matrix [b] = 0 for interior elements. The element coefficient, temporal 
and boundary matrices are given in Appendix B. The element coefficient 
matrix is modified to account for prescribed nodal boundary conditions 
by eliminating the row and column corresponding to that nodal unknown. 
For those boundary nodes at which no boundary conditions (i.e. 
concentrations or fluxes) are prescribed, the normal concentration flux 
across that node is set equal to zero. 

Next, the element matrix differential equations (Eq. 4.24) are 
assembled to form the system matrix differential equation 

5{C} 

[K] {C} + [T] + {p} + [B] {c} = 0 (4.25) 

at 



m 

where all the matrices and arrays are the system equivalents of those 
given in Eq. 4.24. 

Rearranging Eq. 4.25 and replacing the partial derivative with a 
finite difference gives 



[T] 

{ + [K] + [B]} {c} + {F} = 0 (4.26) 

At 



Applying a Crank-Nicholson type representation to temporally discretize 
this equation gives 

{—+ 0[K]n+l + 0[B]"-'l} {C}"+1 = {—. [(l-Q)rK]" - 
At At 

(i-e)[B]"}{c}" + eiF}"""^ + {i-e){F}" (4.27) 



where 0 = degree of implicitness (e = 1, fully implicit; e = o, fully 
explicit), and the superscripts n and n+1 indicate the values of the 
arrays and vectors at the current time step (t = nAt) and at the next 
time step (t = (n+l)At), respectively. The value of e is specified by 
the user. For stability reasons, 9 should be greater than or equal to 
0.50. Using the specified initial and boundary conditions, Eq. 4.27 is 
solved for the NP-NBC unknown nodal concentrations at t = (n+1) At, where 
HP = number of nodes in the system and NBC = number of boundary nodes 
with specified boundary conditions. The method used to solve Eq. 4.27 
is discussed next. 



201 

4.4.4. Equation Solvers 

CSTM-H contains two algorithms which solve Eq. 4.27. One algorithm 
uses the Gaussian elimination technique and the other uses the frontal 
solution program for unsymmetric matrices developed by Hood (1976). The 
frontal algorithm was developed specifically for applications of the 
finite element method to boundary value problems. Although it is based 
on the Gaussian elimination technique, it has advantages over the 
conventional banded matrix techniques in that computer storage 
requirements and computation times may be considerably reduced in 
certain applications. This is especially true for large systems which 
might have 1000 or more variables. For small systems (e.g. with 200 or 
less nodes), there is no appreciable difference in the computation time 
required by the two algorithms. The user of CSTM-H specifies whether 
the band or frontal method is to be used in solving Eq. 4.27. It is 
necessary to create a temporary data file on a disk when the frontal 
algorithm is used. 

4.5. Convergence and Stability 

The accuracy of the numerical scheme used in CSTM-H has been 
investigated in detail by Ariathurai (1974) and Ariathurai et al. 
(1977). These authors reported that rapid convergence to the exact 
analytical solutions was achieved for the numerical formulation for the 
one-dimensional, transient heat conduction problem with and without 
radiation, the one-dimensional, steady-state and transient convection- 
diffusion problem, and for the two-dimensional Laplace equation. 

The results from these convergence tests also indicated that the 
combination of the unconditionally stable finite element formulation 



202 

used to solve the spatial problem and the unconditionally stable Crank- 
Nicholson type finite difference formulation used to solve the temporal 
problem is as well unconditionally stable. However, these tests 
revealed that instabilities might still occur when the Peclet numbers 
(ratio of advection to dispersion, i.e. ul/d^, where u = flow velocity, 

= dispersion coefficient and L = system longitudinal dimension) 
become either too large (greater than = 100) or too small (less than - 
10' ). For too large Peclet numbers, smaller time-steps must be used to 
improve the accuracy of the numerical scheme (Ariathurai et a1 . , 
1977). Too small Peclet numbers rarely ever occur for typical flow 
conditions in estuaries, and therefore associated roundoff errors, which 
can lead to instabilities, should never be a problem in modeling such 
systems. However, spurious results caused by roundoff errors were 
encountered in simulating laboratory depositional experiments with CSTM- 
H. This problem was eliminated by using double precision arithmetic in 
the model. No instability problems were encountered in modeling a 
prototype system with CSTM-H using single precision arithmetic. 

A value between 0.5 and 1.0 should be used for the degree of 
implicitness in order to insure a stable numerical scheme in time. High 
values of this parameter result in a smoother, though no more accurate 
solution than values near 0.5. For modeling tidal bodies of water, a 
time step of the order 10-30 minutes should be used. The element sizes 
should be chosen such that the required detail is obtained in critical 
areas of concern. 



CHAPTER V 
MODEL VERIFICATION AND APPLICATION 



5.1. Introductory Note 

The purpose of this chapter is twofold: 1) to verify the CSTM-H 
model by demonstrating it's capability of predicting cohesive sediment 
transport processes, and 2) to apply the model to a two-dimensional, 
prototype scale body of water. The first objective is achieved by using 
CSTM-H to simulate five different laboratory sediment transport 
experiments and comparing the measured and predicted results. The 
second objective is achieved by using the model to simulate 
sedimentation in a coastal marina. 

5.2. Laboratory Experiments 

A total of four laboratory experiments were conducted at the 
Coastal Engineering Laboratory during this study. Three of the four 
were conducted in a 18.3 m long, 0.61 m wide and 0.91 m deep 
recirculating flume and the fourth in a 0.2 m wide, 0.46 m deep and 0.76 
m mean radius rotating annular flume. The experiments in the 
recirculating and annular flumes are described in Sections 5.2.1 and 
5.2.2, respectively. The results of the model simulations of these 
experiments are presented in Section 5.2.3. 



203 



204 

5.2.1. Recirculating Flume Experiments 
5. 2.1. a. Facilities 

The 18.3 m long flume in which three erosion-deposition experiments 
were conducted is shown in Fig. 5.1. A schematic diagram of this flume 
is given in Fig. 5.2. The main components of the flume are the 
fol lowing: 

1) A chamber located at the upstream end of the flume into which water 
from the recirculating pipe is discharged. A flow straightening 
device was placed in the entry chamber just downstream of the water 
discharge pipe. 

2) The flume has a steel plated bottom and back wall with glass panels 
along the front side. Two electric jacks located 1.5 m downstream 
of the entry chamber can be used to tilt the flume to a maximum 
slope of 0.02. 

3) An underflow tail gate at the downstream end of the flume. 

4) A 1.83 m long, 0.91 m wide and 1.2 m deep transition tank, into 
which the water in the flume flows. 

5) A centrifugal pump with a maximum capacity of 0.164 m^/s. 

6) A discharge control valve. 

7) A 0.2 m diameter, 15.3 m long PVC return pipe. 

8) A 0.33 m wide, 4.9 m long width restricting apparatus was placed in 
the upstream half of the flume along the back wall (Fig. 5.1). The 
central constant width section was 1.52 m in length, and the two 
curved end sections were 1.69 m in length. This apparatus was 
constructed out of 0.31 cm plywood. Two coats of fiberglass resin 
were applied to the bottom and front and back sides to minimize 
swelling of the plywood. Concrete blocks were used for ballast 




Fig. 5.1. Downstream View of Recirculation Flume. Width 
Reducing Device is Shown on Right Side of Flume. 



206 




Q. aj 



c 

> 



c 
o 



c 
o 



Q, OJ w O 

3 a m "w o 9- 
Q- > cr o q: a. 



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207 

(see Fig. 5.1), and silicon and duct tape were placed along the 
submerged edges of the apparatus to prevent sediment from seeping 
behind and/or underneath the apparatus. This width restricting 
apparatus was placed in the flume in order to achieve a region of 
higher flow velocities. 

A 12 mm diameter, 0.9 m long PVC pipe, capped at both ends, with 
1.5 mm diameter holes 5 cm apart, and connected by a rubber hose to 
an air compressor, was placed width-wise across the bottom of the 
transition tank. The turbulence generated by the jets of 
compressed air from this pipe served to increase the turbulence in 
the tank. This helped to minimize the possibility of any sediment 
depositing in the tank, which had a cross-sectional area 
approximately three times larger than that of the flume, and 
therefore a much lower shear stress. 



5.2.1.b. Instrumentation 

Velocity measurements were made during the experiments using a Kent 
miniflow current meter (Model Number 265). The current meter consists 
of a 1 cm diameter, 5 bladed impeller attached to a circular frame, 
which is itself connected to a 48 cm long shaft (Fig. 5.3). There is a 
0.1 mm clearance between the tip of each blade and the base of the 
shaft. Inside the shaft is an insulated gold wire. An electrical 
impedance between the gold wire and the shaft is changed by the rotation 
of the impeller in a conductive fluid (Wang, 1983). This change in the 
impedance modulates a 15kHz carrier signal, the strength of which is 
indicated on the needle dial on the monitor box. The variance in the 
impedance is a function of the impeller's rotation rate, which is in 



208 




Fig. 5.3. Kent Mini-Flow Current Meter. 




Calibration from Supplier 



n I I ! \ ! \ 1 ! ! ^ ! 

^0 2.0 4.0 6.0 8.0 10.0 

INDICATOR READING (Hz) 
Fig. 5.4. Calibration of Kent Mini-Flow Current Meter. 



209 

turn a function of the fluid's flow velocity. The current meter was 
calibrated by Dixit (1982) in the Department of Civil Engineering 
Hydraulic Laboratory's flume and by the author in the rotating annular 
flume at the Coastal Engineering Laboratory. Figure 5.4 shows the good 
agreement obtained between the calibration provided by the manufacturer 
and those obtained by Dixit and the author. 

The current meter shaft was clamped to a vertically positioned 
point gage, which in turn was bolted to a horizontally positioned point 
gage (Fig. 5.5). As seen in this figure, the horizontal point gage was 
bolted to one side of a wooden cart. This set-up permitted exact 
(within the 1 mm accuracy of the two verniers) horizontal and vertical 
positioning of the current meter inside the flume. The wooden cart was 
placed on top of the two 2.5 cm diameter stainless steel rods which 
spanned the length of the flume. The cart was moved on top of the rods 
to the desired location along the longitudinal axis of the flume. 

The time-averaged elevation of the water surface above the flume 
bed was measured during the course of each experiment using the water 
surface elevation measuring device shown in Figures 5.6 and 5.7. This 
device, which operates on the siphon principle, consists of an electric 
point gage, a 5.0 cm diameter plastic tube, two valves, and two 0.5 cm 
diameter clear rubber hoses. The point gage was attached to the top of 
the tube, as seen in Fig. 5.6, and positioned such that the tip of the 
gage touched the water surface at approximately the tube's center. The 
effect of surface tension between the inner tube wall and the water 
inside the tube did not influence the water surface elevation at the 
tube's center because of the relatively large tube diameter (Wang, 
1983). A 3.7 mm diameter glass tube was inserted in the unattached end 



210 




Fig. 5,5. Instrumentation Cart and Setup of Kent 
Mini-Flow Meter and Two Point Gages. 



211 




5.7. Setup of Water Surface Elevation Measuring Device 
(after Wang, 1983). 



212 

of the rubber hose. The glass tube was then inserted through a board 
located a few inches above the water surface in the flume. This was 
done in order to maintain the vertical positioning of the glass tube. A 
hand operated suction pump was used to start a siphon between the water 
in the tube and in the flume. The slow response time between the water 
levels in the flume and tube, caused by the small diameter and the long 
length of the connecting hose, resulted in a filtering of the high 
frequency, turbulent fluctuations of the water surface. As a result, 
only the time-averaged water surface elevation was measured. 

Two of these devices were used to measure the water surface 
elevation at three locations along the flume. One device continuously 
measured the elevation at a downstream station, while the second device 
monitored the elevation at an upstream station and a station in the 
reduced width section. As the water surface elevation can be measured 
at only one location at a time, the upstream elevation was measured by 
first closing the valve of the middle station hose and then opening the 
valve of the upstream station hose, and vice versa. 

The density profile of the sediment bed in the flume was measured 
using a specially designed apparatus (Parchure, 1980). A sketch of this 
apparatus 1s given in Fig. 5.8. To obtain a core of the bed the 2 cm 
diameter plastic tube is inserted in a vertical position through the 
sediment bed until the flume bottom is reached. Then the 15 cm diameter 
plexiglass cylinder with a sealed bottom is lowered concentrically 
around the plastic tube until it also is positioned on the flume 
bottom. The annular space between the 2.5 cm diameter metal tube and 
the outer wall of the plexiglass cylinder (see Fig. 5.8a) is filled 
with denatured alcohol, to which pieces of dry ice (solid carbon 



213 



(a) SKETCH OF APPARATUS I 



ZZ5 cm 



T 

1 






\ 


Top Cylinder IScmdia. ^ 






i 


1 

i 






! 



Rastic Tubes d various heights, 
0.95 cm dia. glued to Ite 
bottom plate 



7.5 cm 

I 



Bottom n 
Cylinder P 

15cm dia-*| 




Bottom Rate-^ 

(b) SKETCH OF /APPARATUS H 



T 

15 cm 

_L 



-2 cm dia plastic tube 

15 cm dia. plexiglass cylinder 
■2.5 cm dia metal tube 



■Annular space for mixture of alcohol 
and dry ice 



Metal 
Plate" 



Sediment 



Porcelein 
Dish 



Riled with ics ojbes 



H7 



^Piston with Screw Red 



5.8. (a) Apparatus I for Obtaining Sediment Core; 

(b) Apparatus II for Sectioning a Frozen Sediment Core 
(after Parchure, 1980). 



214 

dioxide) are added. The combination of the alcohol and dry ice causes 
the sediment core inside the plastic tube to freeze in approximately 20 
minutes. The plastic tube is then removed and placed horizontally in a 
second apparatus, as shown in Fig. 5.8b. The rectangular box is filled 
with ice cubes in order to keep the sediment core frozen. The piston 
with threaded rod is used to push approximately 5 mm lengths of the 
frozen core out of the plastic tube at a time. The ejected section of 
the core is brought in contact with a metal plate in front of the core 
(see Fig. 5.8b), which causes this 5 mm section of the core to quickly 
melt and drop into the porcelein dish. This process is repeated until 
the entire sediment core has been collected in separate dishes. The 
sediment 1n each dish is oven dried and weighed in order to determine 
the dry sediment mass in each 5 mm section of the frozen core. The 
method used to determine the density profile is described in the next 
section. 

Samples of the sediment-water mixture were collected at periodic 
intervals throughout the experiments using the water sampling device 
shown in Fig. 5.9. The horizontal sampling tubes have a 3.2 mm diameter 
and are spaced 4.0 cm apart. This device works on the siphon principle, 
so that when a sample is desired, the end of the rubber hose is lowered 
below the level of the horizontal sampling tube in the flume and 
approximately a 60 ml sample of the suspension is collected in a sample 
bottle. The volume of water contained in the hose is drained into 
another container, and then the 60 ml sample is collected in the 
bottle. This procedure was used so that a sample of the suspension 
which existed in the flume at the sampling time was collected, and not 
that at the previous sampling time. In all the experiments, two samples 
were collected simultaneously from the two lowest sampling tubes. 



215 




Fig. 5.9. Water Sampling Device. 




GRAIN SIZE (mm ) 

Fig. 5.10. Grain Size Distribution of Kaolinite Used for the 
Experiments. 



216 

The suspension concentration of each water sample was determined 
using a Millipore filtering apparatus, an oven, and a Mettler balance 
{Model No. H80) with a ±0.05 mg accuracy. The following procedure was 
used: 

1) Withdraw a certain volume of the suspension from the sample bottle 
using a 10 ml pipette. 

2) Filter this sample through a pre-weighed Millipore filter paper 
with a 0.45 m pore diameter. 

3) Place the filter paper containing sediment in an oven at 50 °C for 
at least two hours. 

4) Weigh the dried filter paper. 

5) Calculate the suspension concentration C = (mass of dried filter 
paper containing sediment - mass of filter paper alone) / (volume 
of suspension filtered). 

Commercial grade kaolinite was the sediment used in the three 
experiments performed in the recirculating flume. The particle size 
distribution of the kaolinite, determined by a standard hydrometer test, 
is shown in Fig. 5.10. The median particle diameter, as seen in Fig. 
5.10, was 1 \im. The CEC value for the kaolinite given by the suppliers, 
the Feldspar Corporation, Edgar, Florida, was 5.2-6.5 meq / lOOgm 
(Dixit. 1982). Tap water was used as the fluid in these experiments. 
The chemical composition of this water is given in Table 5.1. A two 
week period was used to equilibrate the tap water and about 75 kg of 
kaolinite before the first experiment was conducted. 



217 
Table 5.1 

Chemical Composition of the Tap Water (after Dixit, 1982) 



CI 


26 


ppm 


N03 


0.07 


ppm 


Fe 


0.5 


ppm 


K 


1.4 


ppm 


Ca 


25 


ppm 


Mg 


16 


ppm 


Na 


10 


ppm 


Total Salts 


278 


ppm 


pH 


8.5 





5.2. I.e. Procedure 

A zero bed slope was used in all three experiments. The water 
surface elevation measuring devices at the upstream and downstream 
stations were used to adjust the slope to zero. 

The following procedure was used to form a flow-deposited sediment 
bed in each of the experiments. The flume pump was started and the 75 
(dry mass) of sediment was mixed at a shear stress of approximately 0.5 
N/m for four hours. The flume calibration performed by Dixit (1982) 
was used to estimate the shear stress. The mixing was artificially 
enhanced by pushing a rubber wedge along the bottom of both the flume 
and transition tank in order to initially suspend all the sediment. 

After four hours of mixing, the flow in the flume was reduced to a 
shear stress of approximately 0.025 N/m^, which was maintained for eight 



218 

hours. Most of the sediment deposited during this deposition period. 
The non-uniform flow in the flume, which was caused by the width- 
reducing device and the flow under the downstream tail gate, resulted in 
a bed of variable thickness along the length of the flume. 

After the eight hour deposition period, the flow was stopped 
completely. The flow deoosited bed was allowed to undergo self-weight 
consolidation for the following consolidation times: J^^ = 3, 84 and 
240 hours for Test No. 1, Test No. 2 and Test No. 3, respectively. 

At the end of the consolidation period, the flume pump was started 
and the flow rate was slowly adjusted (i.e. increased) so that after 15 
minutes the shear stress in the flume was approximately 0.02 N/m^. This 
same flow rate was maintained for an additional 45 minutes. This 
procedure was followed in order to resuspend the sediment which 
deposited during the consolidation period in the recirculating pipe, 
which has a flow cross-section approximately four times smaller than the 
flume and therefore a shear stress approximately 16 times greater than 
that in the flume, without resuspending any of the sediment in the 
flume. The suspension concentration which existed in the flume after 
this one hour period was regarded as the initial concentration, Cq, at 
the start of each experiment. was determined by water samples 
collected at the end of the one hour resuspension period. In addition, 
the bed surface elevation which existed at the end of the one hour 
period was measured through the front glass panels every 0.91 m along 
the flume test section. These measurements were used as initial bed 
conditions in the model simulations. 

The approximate shear stress history for the three experiments are 
shown in Fig. 5.11. This figure may be interpreted as follows. In Test 



219 



C.I5f 
0 

0.05 

on 



1 \ 

^b2 



r. 



b! 



■^0 



J L 



I I I 



2.0 4.0 6.0 

TIME (hours) 
(a) TEST NO. I 



80 



0.2- 



CO 
00 
LU 
QC 
t- 

cn 

en 
< 



0.1 



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TIME (rxDurs) 

(b) TEST NO. 2 



80 



0.2i — -^FT^ \ \ \ r 



^b. 



J L 



2.0 4.0 6.0 

TIME (hours) 
(c) TEST NO. 3 



80 



ICO 



Fig. 5.11. Shear Stress History for Experiments in the Recirculating 
Flume. 



220 

^ ^'''dc ^ hours), the approximate shear stress in the unrestricted 
width sections of the flume was equal to 0.06 N/m^ for the first two 
hours, then it was increased to 0.12 N/m^ for the third and fourth 
hours, and finally it was decreased to 0.033 N/m^ for the final five 
hours. In Test No. 2 (T^^ = 84 hours), the approximate shear stress was 
0.075 N/m^ for the first 3.5 hours, 0.17 N/m^ for the next 2.5 hours, 
and 0.026 N/m^ for the final five hours. In Test No. 3 (T^^ = 240 
hours), the approximate shear stress was 0.17 N/m^ for the first two 
hours, 0.026 N/m^ for the next five hours, and 0.075 N/m^ for the final 
two hours. It took approximately two minutes to change the flow rate in 
the flume to the new shear stress value and establish steady flow 
conditions. 

During each of the constant shear stress time intervals, water 
samples were collected at sampling Station A (see Fig. 5.12) from the 
two lowest sampling tubes at 0, 2, 5, 10, 20, 30, 40, 50, 60, 75, 90, 
105, 120 minutes, etc. after the shear stress was changed. In order to 
determine if any spatial (i.e. longitudinal and transverse) variability 
in the suspension concentration occurred during the experiments, water 
samples were also taken at alternate sampling times at Station B during 
Test No. 2 and at Station C during Test No. 3. Figure 5.12 shows the 
reach of the flume used in each experiment, and the locations of the 
three water sampling stations (A, B, C), the three water surface 
elevation measuring stations (D, E, F) and the three velocity measuring 
stations (G, H, J). 

The water surface elevation was monitored periodically at each of 
the three measuring stations during each constant shear stress time 
interval. The bed thickness was measured every 0.91 m along the length 



221 




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O 




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CD £_ 
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222 

of the flume's test reach at least once during each constant shear 
stress time interval. In addition, the water temperature was measured 
every hour over the duration of each experiment. 

The vertical velocity profiles at five lateral positions at 
Stations G and J, and at three lateral positions at Station H (see Fig. 
5.12) were measured once during each constant shear stress interval 
using the Kent mini flow meter. 

After the three experiments were completed, the identical bed 
formation process was again repeated. A frozen core of the sediment was 
obtained for the purpose of determining the density profile at each of 
the consolidation times (i.e. 3, 84 and 240 hours) used in the 
experiments. These cores were collected downstream of the width 
restricting device, where the bed had approximately uniform thickness. 

The following procedure was used to determine the bed density 
profile from each segmented core. The density of each 5 mm thick layer 
was determined by dividing the dry sediment mass in that layer by the 
volume. Freezing of the core caused the sediment sample inside the core 
to swell. A uniform correction for the swelling was made by dividing 
the total thickness of the frozen core by the total thickness of the 
original (i.e. before freezing) core, and then dividing the thickness of 
each frozen layer (=5 mm) by this factor. This gives the mean density 
in each layer, from which the density profile may be constructed. 

5.2. l.d. Results 

Figures 5.13-5.15 show the variation of the suspension 
concentration with time during Tests No. 1, 2 and 3, respectively. The 
concentrations determined for both the lower and upper sampling tubes 



223 




0.0 2.0 4.0 6.0 8.0 

TIME (hrs) 



Fig. 5.13. Measured and Predicted Suspended Sediment Concentrations 
for Test No. 1. 



224 




00' I 1 I I I I I I I I 

0.0 2.0 40 6.0 8.0 10.0 

TIME (hrs) 



Fig. 5.14. Measured and Predicted Suspended Sediment Concentrations 
for Test No. 2. 



225 



6.0 



I 4.0 
tr 

H 
UJ 

u 
z 

o 
u 



o 

CO 
UJ 

a. 

3 
C/) 



2.0 




Station 

• A- Upper 
^ A - Lower 
o C- Upper 
^ C - Lower 



n 1 r 

Test No. 3 
V 240 hrs 



Concentrations 
— Predicted 



X 

o 



Measured 



'•A o ^ 




00 



0.0 



2.0 



4.0 
TIME (hrs) 



6.0 



8.0 



Fig. 5.15. Measured and Predicted Suspended Sediment Concentrations 
for Test No. 3. 



226 

are shown in these figures. In addition. Fig. 5.14 shows the suspension 
concentrations determined for the downstream collection station (Station 
B) during Test No. 2, and Fig. 5.15 shows the measured concentrations at 
Station C during Test No. 3. As evident in these two figures, the 
suspension concentration varied slightly in the flume in both the 
longitudinal and lateral directions. The lateral gradient in C was 
caused by the non-uniform lateral velocity profile which existed in the 
flume. The longitudinal gradient was caused by the combined effects of 
longitudinal dispersion, erosion and/or deposition which occurred along 
the length of the flume. The concentration values determined from the 
lower sampling tubes were plotted in Figs. 5.13-5.15 only when they 
differed from the values at the upper sampling tubes by more than ±0.03 
g/1. 

No significant difference (i.e. vertical gradient) in the 
suspension concentration was observed between the upper and lower 
sampling locations during the three experiments, except for the time 
period immediately following changes in the flow rate. This phenomenon 
is best exemplified by the large vertical variation in C (up to 2 g/1) 
during the first half hour period of deposition (from 6.0 to 6.5 hours) 
in Test No. 2 (see Fig. 5.14). Differential settling is the probable 
cause for this observed vertical concentration gradient during the 
initial stages of deposition. The high vertical variation in C (up to 
approximately 1 g/1) found, for example, during the first half hour 
period of erosion (from 2.0 to 2.5 hours) in Test No. 1 (see Fig. 5.13) 
is probably due to higher resuspension rates than vertical diffusion 
rates during this initial period. As the resuspension rate decreases 
due to increasingly smaller values of the excess bed shear (caused by 



227 

the Increase of the bed shear strength with depth below the Initial bed 
surface), the continuing vertical diffusion of sediment reduces the 
magnitude of this vertical concentration gradient, as seen in Fig. 5.13 
from 2.5 to 4.0 hours. 

The depth-averaged velocity at each vertical was determined by 
plotting the vertical velocity profile and then integrating it using a 
planimeter. The water depths at Stations D, E and F were determined by 
subtracting the measured bed thickness from the water surface elevation 
above the bottom of the flume. 

The bed density profiles for the three experiments are given in 
Fig. 5.16. The applicability of the power law relationship between the 
dry sediment density, p, and the depth below the bed surface, z^, as 
given by Eq. 3.76, is apparent for the three measured density profiles. 

5.2.2. Rotating Annular Flume Experiment 

The purpose for conducting an experiment in the rotating flume at 
the Coastal Engineering Laboratory was to verify that the cohesive 
sediment transport model developed during this study can be used to 
predict erosion and deposition rates in an unsteady flow field. 

5. 2. 2. a. Facilities 

The annular flume in which the experiment was performed is shown in 
Fig. 5.17. This flume has the following dimensions: 0.21 m wide, 0.45 
m deep and 0.76 m mean radius. The flume consists of three main 
components: 1) a rotating circular fiberglass channel which holds the 
sediment-water mixture, 2) an annular ring with a slightly smaller width 
and the same mean radius as the channel, and 3) a steel frame and 



228 





p / p 



. 5.15. Measured Bed Density Profiles for Experiments in 
Recirculating Flume. 



229 




230 

electric motors. The ring, positioned in contact with the water 
surface, and the channel are rotated simultaneously in opposite 
directions in order to achieve a nearly uniform turbulent shear field in 
the channel, and to minimize the effects of rotation-induced radial 
secondary currents. This design and operational procedure eliminates 
the need for aggregate-disrupting elements such as circulatory pumps, in 
which very high shearing rates usually occur. The required bed shear 
stress is attained by adjustment of the rotational speeds of the channel 
and the ring. Four taps, located on the outer channel wall (see Fig. 
5.17), are used to collect suspended sediment samples from the channel. 

5.2.2.b. Instrumentation 

A Hewlett Packard HP-85 micro-computer with two digital-to-analog 
converter units were used to control the rotation rates and 
accelerations of the channel and the ring. The micro-computer was 
programmed to generate the desired flow field in the channel. 

The bottom sediment from Lake Francis, Nebraska and tap water with 
a 10 ppt solution of commercial grade sodium chloride were used in this 
experiment. X-ray diffraction analysis performed by the Soil 
Characterization Laboratory at the University of Florida revealed that 
this sediment is predominantly composed of montmorillonite, illite, 
kaolinite and quartz. 

5.2.2.C. Procedure 

The sediment and 10 ppt saline water were placed in the channel. A 
30.5 cm depth of flow was used. The sediment-water mixture was mixed at 
a shear stress of approximately 1.7 N/m^ for 24 hours. The flume was 



231 

then stopped and the suspended sediment was allowed to deposit and 
undergo self-weight consolidation for 40 hours. 

The HP-85 micro-computer was programmed to generate a uni- 
directional, semi-diurnal, constant depth tidal flow. The cross- 
sectionally averaged sinusoidal velocity-time record used in this tidal 
cycle experiment is given in Fig. 5.18. Because of a mechanical probl 
with the bearings in the channel's drive shaft, it was not possible to 
have a true slack period in this experiment. Though not apparent in 
Fig. 5.18, the flow velocity was discretized into five minute 
increments, during which the velocity varied as follows: during the 
first 30 seconds of each five minute time increment, the velocity was 
linearly increased or decreased to the next value (as determined by a 
sinusoidal velocity-time relationship), while for the remaining 4.5 
minutes, a constant velocity was maintained. 

The program was started and run for two tidal cycles (25 hours) in 
order to establish quasi -steady state conditions. Water samples were 
collected from the mid-depth tap at the end of this 25 hours and every 
five minutes thereafter for 16 hours. The samples were collected just 
before the end of each five minute increment (i.e. before the velocity 
was changed). 

The temperature of the water in the channel was measured every 
hour. In addition, water was added every hour to replace that withdrawn 
in the water samples in order to keep the ring in contact with the water 
surface. 



em 



232 




233 

5.2.2. d. Results 

Figure 5.18 shows the variation of the suspension concentration 
over the duration of the 16 hour experiment. Also plotted in this 
figure is the variation of the cross-sectionally averaged velocity. A 
short lag between the flow velocity and the suspension concentration is 
seen to occur immediately following the occurrences of minimum and 
maximum velocities. 

5.2.3. Model Simulations 

The test reach of the recirculating flume was divided into the 34 
element, 141 node finite element grid shown in Fig. 5.19. Zero sediment 
flux boundary conditions were used for the upstream and downstream flow 
boundaries. The depth-averaged velocities measured at Stations G, H and 
J were used for the nodal velocities in elements 1-2, 7-10, and 15-34, 
respectively. The nodal velocity vectors in the converging section 
(elements 3-6) of the flume and in the diverging section (elements 11- 
14) were determined from continuity considerations. Likewise, the water 
surface elevations measured at Stations D, E and F were used for the 
nodal values in elements 1-2, 7-10, and 15-34, respectively. The nodal 
water surface elevations in elements 3-6 and 11-14 were determined by 
linear interpolation of the values measured at Stations D and E, and 
Stations E and F, respectively. The initial bed thicknesses measured 
every 0.91 m along the flume were used as the initial bed thicknesses 
for the even-numbered elements. The initial bed thickness of the ith_ 
odd-numbered element was assumed to be equal to that of the (i+l)th 
even-numbered element, for i=l, 33. The dry mass of the sediment 
forming the bed in each element at the start of each experiment was 



234 



(r 






5^ 


6 




7 


8 




9 


10 




II / 


12 




/,4 



15 


16 


17 


IB 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 


29 


30 


31 


32 


33 


34 



Fig. 5.19. Finite Element Grid of Recirculating Flume; Distorted 
Sketch - Width: Length = 4.1: 1.0. 



235 

determined using the measured bed thickness and measured bed density 
profile. The dry mass in each element and the measured bed density 
profile were read into CSTM-H in order to form the initial bed for each 
experiment. The parameters which characterize the erosional and 
depositional characteristics of kaolinite in tap water, determined by 
Dixit (1982) and Mehta (1973), respectively, and the consolidation 
parameters given in Chapter III, Section 3.7.1 were used in CSTM-H to 
simulate the three experiments in the recirculating flume. The 
dispersion coefficients were calculated using the dispersion 
algorithm. A two minute time step was used in the model simulations. 

Comparisons of the predicted and measured suspension concentrations 
for the three experiments are shown in Figs. 5.13-5.15. Satisfactory to 
good agreement is observed in all three experiments. A discussion of 
these results is given in Section 5.4. 

A four element, 23 node straight grid was used to represent the 
annular flume in simulating the tidal cycle experiment. The length of 
this grid system was set equal to the circumference of a circle with a 
radius equal to the mean radius of the flume. The suspension 
concentration at the "downstream" flow boundary was used for the 
"upstream" boundary condition in order to represent they were the same 
boundary. The velocity record shown in Fig. 5.18 and the erosional and 
depositional characteristics of Lake Francis sediment reported by Mehta 
et al_. (1982a) and Hayter and Mehta (1982) were used in CSTM-H. A 2.5 
minute time step was used in the model simulation. 

A comparison of the predicted and measured suspension 
concentrations is as well shown in Fig. 5.18. A discussion of the 
simulation result is given in Section 5.4.. 



236 

5.3. Simulation of WES Deposition Experiment 

A deposition experiment conducted in the 99.7 m long, 0.46 m deep 
and 0.23 m wide non-recirculating flume at the U.S. Army Corps of 
Engineers Waterways Experiment Station, Vicksburg, Mississippi was 
simulated with CSTM-H. A schematic diagram of this flume is shown in 
Fig. 5.20. A detailed description of the flume is given by Dixit etal. 
(1982). The purpose of this experiment and three other experiments 
conducted in the 99.7 m long flume was to investigate the phenomenon of 
sediment sorting in the longitudinal direction, the occurrence of which 
is well documented in muddy estuaries (Edzwald et al_. , 1974; Dixit et 
al_'. 1982). 

The sediment used in this experiment was the commercial grade 
kaolinite described in Chapter III. The fluid used was tap water with a 
chloride concentration of 18 ppm. pH = 7.8 and sodium adsorption ratio 
SAR = 2.07 (Dixit etal., 1982). 

The procedure used in this deposition experiment was the 
following. A 100 g/1 sediment-water slurry was prepared by mixing the 
kaolinite and water for three hours. The slurry was injected at a 
specified rate into tap water which flowed from the headbay into the 
flume (see Fig. 5.20). The slurry injection rate was regulated so that 
the resulting suspension had a concentration of approximately 4.0 g/1. 
The flow rate was held constant for three hours, after which only the 
clear water flow was maintained until the sediment cloud had passed out 
of the flume. The thickness of the deposited bed was measured 15 
minutes after the end of the experiment. Vertical and horizontal 
velocity profiles and the water surface elevations were measured at 
several stations along the length of the flume. In addition, vertical 



237 




= 1 

a> ^ o a i_ p 



(U 
3 



o 
o 



0) 



B 
X 

Q. 
3 

f 

+J 
0) 
00 



o ^ 

CM 
C CO 

o cn 

•1— t-H 

■M 

fC 

+J . 

C "—I 

<D fdl 
s- (d| 

Q- 

q; -1- 

X 
o •■- 
•I- Q 

+-> 

ra S- 

E OJ 
CD +J 

J=. <+- 
O (C 

CO > 



o 

CNJ 



238 

suspension concentration profiles were measured at several stations 
along the centerline of the flume every 15 minutes during the course of 
the experiment. 

The mean flow depth, mean velocity and the bed shear stress during 
the experiment were 0.162 m, 0.091 m/s, and 0.033 N/m^, respectively 
(Dixit etal_., 1982). 

This experiment was simulated using a 10 element, 53 node grid and 
a four minute time step. The stated flow conditions, the depositional 
properties of kaolinite in tap water found by Mehta (1973) and the 
consolidation properties of kaolinite given in Chapter III, Section 
3.7.2 were used in CSTM-H to perform the simulation. For the upstream 
boundary condition, the suspension concentration was set equal to 4.0 
g/1 for three hours, and then set to 0.0 g/1 for the remainder of the 
experiment. A zero concentration flux was used for the downstream 
boundary condition. The model simulation was continued until the 
maximum predicted concentration in the flume was 0.04 g/1 (1% of the 
initial injection concentration). Figure 5.21 shows the comparison 
between the measured and predicted bed thickness profile at the end of 
the experiment. A discussion of the results of this simulation is given 
in the next section. 

5.4. Discussion of Results 

The model simulations of the three experiments in the recirculating 
flume yielded good agreement between the measured and predicted temporal 
variations in the suspended sediment concentration, as seen in Figs. 
5.13-5.15. In Test No. 1, it is apparent from the measured 
concentration-time record that during the two erosion intervals (i.e. 



239 



QJ 



-o 
> 

CD 

in 

o 



o 



O 

00 



O 



o 
to 



o 

IT) 



o 

ro 



8 



OJ 



00 tD O 

— d o d 
(ujo) SS3N>iDIHl 039 



240 

during the first four hours) the bed was primarily redispersed (mass 
eroded) by the flow-induced bed shear stress. Deviations between the 
measured and the predicted suspension concentrations (i.e. between the 
observed and the predicted rates of erosion) occurred because the 
consolidation algorithm in CSTM-H stipulated that the bed, for which T^^ 
> 2 hours, would undergo the slower process of resuspension when 
subjected to an excess shear. Even though the predicted erosion rate 
during the first hour of each erosion interval was less than that 
reflected in Fig. 5.13, good agreement was achieved between the total 
mass of sediment that was measured and predicted to erode over the 
duration of each interval. Nevertheless, the fact that the bed was 
apparently redispersed when subjected to the given bed shear stress 
points out a limitation of the erosion-consolidation algorithm in CSTM-H 
and an area for future research. 

The deposition stage in Test No. 1 was in Range IB (see Fig. 3.48) 
where the settling velocity decreases with a decrease in the suspension 
concentration according to W3 a c\ with n = 1.33. The good agreement 
obtained between the measured and the predicted concentrations in this 
interval (see Fig. 5.13) was obtained by adjusting the value of the 
proportionality constant (kJ in Eq. 3.51b) between and c". 

In Test No. 2 with T^^ = 84 hours, the measured suspended sediment 
concentrations again indicate that the top layer of the bed was 
apparently redispersed during the first erosion step (see Fig. 5.14). A 
slightly better agreement between the measured and the predicted 
concentration was achieved during this first step by increasing the 
value of the parameter a (see Eq. 3.13) in the top bed layer from the 
value obtained by Dixit (1982). 



241 

The measured concentrations during the second erosion interval in 
Test No. 2 indicate that the bed eroded aggregate by aggregate 
(resuspended) after the initial five minute interval (from 3.50 hours to 
3.58 hours), during which redispersion occurred. Because the bed was 
resuspended during all but the first five minutes of this interval, a 
better agreement was obtained between the measured and the predicted 
concentrations, and therefore also between the actual and the predicted 
rates of erosion. However, as apparent in Fig. 5.14, the predicted 
concentrations were consistently lower than the measured concentrations 
during the second erosion interval. A better agreement could have been 
obtained by increasing the values of the aggregate erosion rate (see 
Eq. 3.13). 

Good agreement was again achieved during the deposition stage in 
Test No. 2, although the predicted deposition rate was slightly higher 
than that observed during the last two hours of the experiment. This 
resulted in a lower predicted than observed concentration at the end of 
the experiment. 

In Test No. 3 with T^^ = 240 hours, resuspension occurred during 
the first erosion interval after the first 10 minutes during which 
redispersion apparently occurred. The redispersion resulted in higher 
observed than predicted suspension concentrations for the first hour. 
Fairly good agreement was achieved during the second hour. 

Good agreement was obtained between the measured and the predicted 
concentrations during the five hour period of deposition in Test No. 3 
(see Fig. 5.15), although from 2.5 hours to 5.0 hours the predicted 
deposition rates were slightly less than the observed rates, while 
during the last hour of the deposition interval (from 6.0 hours to 7.0 



242 

hours) the predicted deposition rates were slightly greater than the 
observed rates. 

The measured suspension concentrations during the second erosion 
step in Test No. 3 (from 7.0 hours to 9.0 hours) indicate that the top 
portion of the sediment which deposited during the previous five hours 
was eroded in mass by the increased bed shear stress. Redispersion was 
simulated by CSTM-H to occur during the first time step of increased bed 
shear. The occurrence of redispersion is not evident in Fig. 5-15 
because of the extremely thin unconsolidated new deposit (UND) layer 
(and therefore small quantity of dry sediment mass which forms this 
layer) used in this simulation. The subsequent resuspension predicted 
by CSTM-H yields a slightly greater eroded sediment mass than measured. 

Figure 5.18 shows the comparison between the model simulation of 
the tidal cycle experiment in the rotating flume and the measured 
suspension concentrations. A good agreement between the predicted and 
measured concentrations is seen. The most noteworthy differences are 
that the measured concentrations lag the predicted values by 
approximately 20 minutes at the times of maximum concentrations (which 
correspond to the times of maximum or peak tidal flow velocities), while 
the predicted concentrations lag the measured values by approximately 20 
minutes at the times of minimum concentrations (which corresponded to 
the times of minimum flow velocities). Thus, assuming the velocity-time 
record used in the simulation is correct, the predicted periods of 
erosion and deposition are approximately 40 minutes longer and shorter 
than, respectively, the observed periods. There are at least four 
possible explanations for this difference in the predicted and observed 
periods of erosion and deposition. 1) A time lag between the change in 



243 

the rotation rates of the ring and the channel of the rotating annular 
flume and the resulting response in the flow velocity. 2) The 
occurrence of deposition even in an accelerating flow when the flow 
velocity is below a certain minimum critical value. 3) The decrease in 
the flow acceleration leading up to a maximum velocity might cause a 
portion of the suspended sediment in the upper half of the water column 
to settle below the mid-depth tap where the water samples were 
obtained. This would result in a decrease in the suspension 
concentration as determined from the mid-depth sampling location, and 
therefore seem to indicate that deposition had occurred, even though the 
sediment might have just settled below mid-depth and not have actually 
deposited on the bed. 4) The 20 minute lag between the onset of 
accelerating flow and the increase in the measured suspension 
concentration (as observed at about 6.0 hours in Fig. 5.18) might have 
been caused by the lag time between the erosion of sediment and the 
vertical diffusion of this sediment up to the mid-depth sampling 
location. The merits of the second and fourth possible explanations are 
questionable in light of the fact that no time lag is observed between 
the measured and predicted concentrations at 12.5 hours in Fig. 5.18. 
The first possible explanation is not very plausible since the response 
time of the water in the channel to changes in the rotation rates of the 
ring and the channel is generally of the order of one to three 
minutes. The explanation given (number three) for the observed lag 
between the predicted and measured concentrations at maximum flow 
velocities needs to be investigated in the future by repeating this 
experiment and taking water samples at several locations over the flow 
depth during the latter half of accelerating flows. 



244 

As observed in Fig. 5.18, the measured increase in the amount of 
sediment eroded per half tidal cycle (i.e. during the two accelerating 
flow periods per tidal cycle) was predicted fairly accurately by CSTM- 
H. The explanation for this slight increase in eroded sediment is the 
following. After the top bed layer is eroded, the shear strength of the 
now exposed lower bed layer decreases as the bed surface swells in 
response to the removal of the overburden pressure. This phenomenon is 
simulated in CSTM-H by changing (decreasing) the bed shear strength at 
the new bed surface to the value of that existed at the bed surface 
at the end of the previous time step. Thus, as the experiment 
continues, the bed shear strength at the various depths (below the 
initial bed surface) to which the bed is eroded becomes slightly less, 
which of course increases the susceptibility of the exposed sediment to 
erosion. This slight decrease of the bed shear strength of the surface 
due to swelling is greater, on a short term basis, than the increase of 
the shear strength due to consolidation. The fact that CSTM-H simulates 
the slight increase in the amount of sediment eroded per time step with 
good accuracy, as seen in Fig. 5.18, is an indication that this 
representation of the decrease in i^ at the bed surface is realistic. 

The simulations of the three experiments in the recirculating flume 
and tidal cycle experiment in the rotating flume have verified the 
predictive capability of CSTM-H. The ability of CSTM-H to model the 
longitudinal sorting process which occurred in the deposition experiment 
in the flume at WES is discussed next. 

As apparent in Fig. 5.21, satisfactory agreement was achieved 
between the measured and predicted deposit thicknesses in the lower 65 m 
reach of the WES flume. The measure thickness in the first 35 m 



245 

possibly shows the influence of longitudinal sorting, which results in a 
variation in the rates of deposition of the suspended aggregates along 
the flume. A causative factor of this phenomenon is thought to be 
differences in the composition of the particles which form the 
aggregates (Dixit etal_., 1982). CSTM-H is not capable of simulating 
longitudinal sorting because the parameters (specifically tgQ and 
which characterize the log-normal depositional law are assumed to be 
spatially invarient. However, if the relationships between t^Q and 
with distance along the flume were known, it would be possible to 
incorporate these into the deposition algorithm, and thereby have the 
capability of predicting the effect of longitudinal sorting on the rates 
of deposition. When CSTM-H is used to predict the sediment movement in 
an estuary, the variation of tgg and in both the x- and y- directions 
would have to be determined. The number of field and laboratory 
experiments that would need to be performed in order to determine the 
relationships tgQ = tgg (x, y, t^) and = a, (x, y, , (or even tgQ 

" ■'^50 '^b^ ^2 " °2 "^b^^ ''^ thought to be impractical. Thus, 
only the relationships t^Q = t^Q [x^) and = (t^) were incorporated 
into CSTM-H. 

Another possible explanation for the smaller measured bed thickness 
in the upstream 35 m reach is the increased turbulence present in the 
flume immediately downstream of the point of injection. Increased 
turbulence would result in lower rates of deposition, and therefore to 
smaller bed thicknesses. It needs to be emphasized that the preceding 
discussion on the discrepancy in the upstream 35 m reach is based on the 
assumption that the bed thickness in this reach is approximately uniform 
and equal to the measured value at 25 m. 



246 

5.5. Model Applications 

The utility of CSTM-H is demonstrated by simulations of the 
sedimentation in Camachee Cove Marina and the suspended sediment 
transport in a 10 km hypothetical canal. Camachee Cove Marina is 
located on the Tolomato Ri ver-Intercoastal Waterway about 150 m north of 
the Vilano Bridge in St. Augustine. Florida. An aerial photograph of 
the basin is shown in Fig. 5.22. The single entrance channel to the 
basin is about 180 m long and 60 m wide, and has naturally sloped 
banks. The surface area of the basin is approximately 33.370 m^. The 
semi-rectangular shaped basin has approximate dimensions of 300 m in 
length and 100 m in width, being tapered towards the south end 
(Srivastava. 1983). The bathymetry of the marina basin determined from 
a survey conducted in September. 1982 is shown in Fig. 5.23. The local 
scour holes seen in this figure were probably caused by boat 
propellers. The bulkhead of the basin is made of concrete sheetpile. 
The sediment material in the basin was found to have eight percent of 
organic matter by weight, and a median particle diameter which varied 
from 10 m in the northern end of the basin to 40 m in the center of 
the basin. Such a distribution is not surprising since the largest 
particles would be expected to deposit in the central wide section of 
the marina. 

The hydrographic and sediment data required to model both the 
predominantly tide-induced circulation and sedimentation in the basin 
were collected by the Coastal Engineering Laboratory at the University 
of Florida. A detailed description of the field study is given by 
Srivastava (1983). The main findings of this hydrographic study were 
the following: 1) The tide is semi-diurnal, with a mean range of 1.4 



247 





Fig. 5.23. Bathymetry of the Entire Basin Obtained in September, 1982. 



249 

m. 2) The maxlinum velocity in the entrance is approximately 2.4 
cm/s. 3) The average suspended sediment concentration was 6.2 tng/1 for 
the months of July and August, 1982. and 6.8 mg/1 for November, 1982 
through July, 1983. 

The hydrodynamic modeling was performed using the two-dimensional, 
depth-averaged finite element flow model RMA2 (Norton et al .. 1973). 
The finite element grid used in modeling both the tidal flow and 
cohesive sediment transport is shown in Fig. 5.24. The results from the 
flow modeling as well as the required sediment data (e.g. suspension 
concentration-time record at the entrance, sediment settling 
characteristics) given by Srivastava (1983) were used in modeling the 
sedimentation in the marina using CSTM-H. The results from this 
modeling effort are shown in Fig. 5.25, which shows contours of the 
predicted amount (thickness) of sediment deposition in centimeters per 
year. The mean depth of the marina is predicted to reduce by 8.6 
cm/year, which is 42% less than the measured 14.8 cm/year (Srivastava, 
1983). The measured sedimentation rate was obtained by comparing 
bathymetric surveys conducted in March, 1980 and September, 1982. The 
observed deposition pattern is not unexpected, as the greatest amount of 
sediment deposition would be expected in the central wide portion of the 
basin where the flow velocities are considerably lower than in the 
relatively narrow channel. Three limitations of this modeling effort, 
and the probable reasons for the difference between the measured and 
predicted deposit thickness, are discussed next. 

The predicted sedimentation rate is representative of fair weather 
conditions only, in that the tide and suspended sediment concentration 
at the marina entrance (which were used for the boundary conditions in 



250 




251 




252 

the flow and sediment transport model, respectively) were measured 
during fair weather. It is believed that storm events may enhance the 
rate of sediment intrusion into the basin. This is attributable to the 
probable increase in concentration of suspended sediments caused by the 
erosion of more sediment by the storm agitated exterior body of water 
(i.e. the Tolomato River). 

The water samples collected over the depth of flow in the entrance 
channel during the field study, from which the depth-averaged suspension 
concentrations were determined, were only collected down to 30 cm above 
the bottom. Thus, the proportion of the suspended load, which may be 
considerable, that is transported into and out of the basin in the 
bottom 10% of the flow depth was not accounted for in the boundary 
conditions used in the cohesive sediment transport model. 

The influence of horizontal circulation due to shear flow at the 
entrance to the marina, as well as that due to wind-induced vertical 
circulation was not accounted for in the flow modeling. Thus, only the 
advective and dispersive transport of suspended sediment due to tidal 
flow in the marina was accounted for in the sediment transport 
modeling. The combination of these three factors, which were not 
accounted for in this modeling effort, are felt to be the reason why the 
predicted sedimentation rate was 42% less than the measured rate. 

The dimensions of a 10 km long hypothetical canal are shown in Fig. 
5.26. The canal was divided into nine elements and 48 nodes, with the 
length of elements 1 - 3 equal to 833 m and that of elements 4 - 9 equal 
to 1250 m. The canal was assumed to have a uniform bottom roughness, as 
quantified by a Mannings coefficient of 0.02, which is a reasonable 
value for a straight natural waterway with a muddy bottom. The depth 



254 



and the mean velocity at nodes 1. 2 and 3 were taken to be 5.0 m and 0.5 
m/s respectively. The velocities and water depths at nodes 4 - 48 were 
evaluated using the conservation of energy and mass equations for an 
open channel. The total drop in the water depth over the 10 km distance 
due to frictional resistance and the gradual enlargement in width at 
element 5 was determined to be 0.16 m. The initial suspension 
concentration in the canal was taken to be 0.0 g/1. The following 
boundary conditions were used: nodes 1. 2 and 3: C(t) = 0, and nodes 
46, 47 and 48: 9c{t)/ax = 0. The upstream (i.e. nodes 1. 2 and 3) 
boundary condition states that no suspended sediment was transported 
into the canal from upstream sources, while the downstream (i.e. nodes 
46, 47 and 48) boundary condition stipulates that the longitudinal flux 
of suspended sediment across the downstream boundary was zero. In 
elements 1 - 4. an initial, partially consolidated Lake Francis sediment 
bed 0.17 m in thickness was assumed to exist, while in elements 5 - 9, 
no initial bed was present. 

Erosion of the initial sediment bed occurred in elements 1 - 4, 
while deposition of the sediment suspended in the first four elements 
occurred in elements 5 - 9. The suspension concentration-time record 
for elements 4 and 5 are shown in Figs. 5.27 and 5.28 for salinities of 
0, 1, 10 and 35 ppt. As evidenced by the over three-fold decrease in 
concentration between Figs. 5.27 and 5.28, a high percentage of the 
suspended sediment deposited in element 5. Also observed in these two 
figures is a reduction in the quality of sediment suspended with 
increase in the salinity. This observation follows from the previously 
described effect of salinity on the rate of erosion. Also apparent is 
the small effect of salinity on the rates of deposition. In conclusion. 



255 




Fig. 5.27. Predicted Suspended Sediment Concentration-time Record 
for Element 4 in Hypothetical Canal. 



256 




TIME (Hrs) 



5.28. Predicted Suspended Sediment Concentration-time Record 
for Element 5 in Hypothetical Canal. 



257 

these simulations demonstrate the significant influence of salinity on 
the transport rate of cohesive sediments for salinities less than about 
10 ppt, and the diminished influence of salinity for salinities greater 
than 10 ppt. 

5.6. Model Limitations 

A two-dimensional, depth-averaged model such as CSTM-H can strictly 
be applied only to estuaries, harbors and basins (such as marinas) where 
the horizontal dimensions of the water body are at least one order of 
magnitude greater than the vertical dimension. Applications to 
partially mixed water bodies or especially to highly stratified water 
bodies should be made when only rough estimates of some sedimentary 
process (e.g. shoaling rate) are required. 

Currently the model has the capability of simulating the movement 
of only one constituent (e.g. cohesive sediment, water temperature, or 
algae, provided the source/sink expressions for a nonconservati ve 
constituent are known). It is possible, however, to modify the model so 
that any number of constituents may be incorporated. 

Probably the main "limitation" of a model arises from three 
sources: 1) insufficient data, 2) poor quality of data and 3) 
limitations of the hydrodynamic modeling. The first two sources are 
attributable to the fact that, owing mainly to time and cost 
consideratins, all the bathymetric, hydraulic and sedimentary data 
required for use in such a model are rarely, if ever, measured and/or 
collected in the body of water being modeled. In addition, the quality 
of the data is often questionable. Data requirements and the field 
collection and laboratory testing programs required to obtain these data 
are briefly described in Appendix D. 



258 

The third source is itself often the result of the first two, 
inasmuch as progress has been achieved in the past two decades in 
modeling estuarial hydrodynamics (Leendertse et al_. , 1973; King et al., 
1973; Liu and Leendertse, 1978). 

The importance of experience in effectively using the model cannot 
be over emphasized. Experience gained through knowledge of the physical 
systems being modeled and repeated applications of the model will 
enhance the user's ability to choose the proper values of the various 
parameters, e.g. time-step size. The user will also gain the ability to 
anticipate the effect of changing the value of a particular parameter by 
a certain percentage on the model solution (i.e. model sensitivity). 

5.7. Model Applicability 
5.7.1. Water Quality Problems 

The model can be used to assist in the performance of the following 
water quality related computational tasks: 

1. ) Assessment of the disposition of dissolved and sorbed pollutants, 

possibly either transported to an estuary or harbor by stormwater 
runoff or released into these water bodies by nearby industries, 
and their effect on the receiving waters and the aquatic ecosystem 
therein, when linked with a particulate contaminant transport model 
that contains a sorption submodel (Onishi and Wise, 1979). 

2. ) Prediction of the effect of reduced sediment inflows to estuaries, 

caused by upstream water storage and subsequent use, to ascertain 
the degree of waste water management required to control estuarial 
water pollution. 



259 

3.) Prediction of the limitation of sunlight penetration in estuarial 
waters resulting from high turbidity levels which, in turn, are 
caused by high concentrations of suspended sediment. This reduced 
light penetration can cause the algae multiplication rate to 
decrease significantly, and thus affect the entire aquatic 
ecosystem. 



5.7.2. Sedimentation Management Problems 

The model can be used as a tool to help solve the following 
sedimentation problems: 

1. ) Prediction of the movement of dredged material released in open 

waters in order to estimate the effect of the disposal at a given 
location in the water body on the shoaling rates elsewhere, and in 
particular in the dredged area. 

2. ) Selection of harbor sites in estuaries and bays where shoaling is 

minimized. 

3. ) Prediction of changes in the sedimentary regime that may occur as a 

result of a proposed change or development of an estuary or harbor, 
such as the dredging of new navigation channels and the possible 
change in the salinity field (e.g. further inland intrusion) caused 
by the proposed change. 

4. ) Estimation of shoaling rates and maintenance dredging requirements 

in areas of very low flow such as marinas, harbors and docks, and 
recommendation of means by which shoaling rates might be minimized. 

5. ) Prediction of the spatial (primarily longitudinal) variance in the 

shoaling and/or erosion rates, caused by varying flow conditions 
and salinities, along the entire reach of an estuary. 



CHAPTER VI 
CONCLUSIONS AND RECOMMENDATIONS 



6.1. Summary and Conclusions 

Cohesive sediments are comprised largely of terrigenous clay-sized 
particles. The remainder may include fine silts, organic matter, waste 
materials and small quantities of very fine sand. The electrochemical 
surface repulsive forces which act on each elementary clay particle are 
approximately six orders of magnitude larger than the gravitational 
force. As a result, the physicochemical properties of cohesive 
sediments are controlled mostly by these surface forces. 

In water with a very low salinity (less than about 1 ppt) the 
elementary particles are usually found in a dispersed state. A slight 
increase in the salinity (up to 2-3 ppt) is sufficient to repress the 
repulsive surface forces between the elementary particles, with the 
result that the particles coagulate to form aggregates. Each aggregate 
may contain thousands of elementary particles. Coagulation depends upon 
interparticle collision and cohesion after collision. The three 
principle mechanisms of interparticle collision in suspension are 
Brownian motion, internal shearing and differential sedimentation. 
Cohesion of elementary particles is caused by the presence of net 
attractive surface forces. The latter condition is caused by the 
increased concentration of dissolved ions, which serves to depress the 
double layer around each particle and allow the attractive forces to 
predominate. 

260 



261 

Most estuaries contain abundant quantities of cohesive sediments 
which usually occur in the coagulated form in various degrees of 
aggregation. The transport of cohesive sediments in estuaries is 
strongly influenced by the coagulation behavior of dispersed sediment 
particles, which is controlled by the chemical composition of the 
suspending fluid, the hydrodynamic conditions, the concentration of 
suspended sediments and the physicochemical properties of the sediment. 

Sediment related problems in estuaries include shoaling in 
navigable waterways and water pollution. The mixing zone between upland 
fresh water and sea water, as well as areas such as dredged cuts, 
navigation channels, harbors and marinas are favorable sites for 
sediment deposition. Since estuaries are often used as transportation 
routes, it is necessary to accurately estimate the amount of dredging 
required to maintain navigable depths in these water bodies. A 
significant portion of the pollution load in a water body is typically 
transported sorbed to cohesive sediments. Therefore, the importance of 
considering the transport of these sediments in predicting the 
disposition of pollutants introduced into an estuary cannot be over 
emphasized. 

A two-dimensional, depth-averaged, finite element cohesive sediment 
transport model, CSTM-H, developed during this study may be used as a 
tool in the field of estuarial management to predict the fate of sorbed 
pollutants and the frequency and quantity of dredging required to 
maintain navigable depths. CSTM-H is a time varying model that is 
capable of predicting the horizontal and temporal variations of the 
depth-averaged suspended sediment concentrations and bed surface 
elevations in an estuary, coastal waterway or river. The two- 



262 

dimensional, depth-averaged advection-dispersion equation with 
appropriate source/sink terms is solved at each time-step for the nodal 
concentrations. Previous models are not as comprehensive as they use 
mathematical descriptions (or algorithms) of the transport processes 
(that are considered), that are based on limited studies conducted prior 
to the early 1970 's. In this study field evidence and the considerable 
amount of experimental research that has been conducted on the mechanics 
of cohesive sediment transport since that time have been used to develop 
new algorithms which describe the processes of erosion, dispersion, 
settling, deposition, bed formation and consolidation. This has 
resulted in a model whose oredictive capability is improved over that of 
previous models. A summary of the algorithms and the improvements 
achieved in the mathematical representations of these transport 
processes is given next. 

Deposited estuarial sediments occur in three different stages of 
consolidation: unconsolidated, partially consolidated and settled 
(fully consolidated). Unconsolidated deposits, referred to as 
stationary suspensions, possess a very high water content and low shear 
strength and are redispersed, or mass eroded, when subjected to an 
excess bed shear stress. Partially consolidated deposits have a 
somewhat lower water content and higher shear strength and are 
resuspended aggregate by aggregate, i.e. undergo resuspension, when 
subjected to an excess shear stress. Settled, or fully consolidated 
beds possess a much lower water content, a much higher shear strength 
and as well are resuspended aggregate by aggregate when subjected to an 
excess shear. The shear strength and the density of partially 
consolidated beds have been shown by laboratory tests to increase with 



263 

depth below the water-bed interface, and as such are vertically 
stratified. Both stationary suspensions and partially consolidated beds 
undergo consolidation due to overburden pressure, with the bed density 
and shear strength increasing with time of consolidation. In settled 
beds, the shear strength and the density profiles exhibit relatively 
uniform properties over the depth. The sediment bed schematization 
incorporated in CSTM-H allows for the above-mentioned three bed 
sections, and divides each section into a characteristic number of 
layers. Within each layer, the bed shear strength and density are 
assumed to vary in a linear manner with depth. The number of layers as 
well as the shear strength and density profiles in each section must be 
determined from laboratory erosion tests. Even though a stationary 
suspension is not a true bed, it is represented as such in order to 
account for the sediment mass which forms this suspension. Previous 
models use a constant bed shear strength and bed density for each layer, 
and use only a single layer for the partially consolidated bed 
section. Therefore, the stratified nature of partially consolidated 
beds is not represented in these models. 

The bed formation algorithm incorporated in CSTM-H uses the assumed 
linear bed density profile in each layer to iteratively solve for the 
thickness of bed formed by the deposition of a given mass of sediment. 
The bed structure (i.e. bed shear strength and density profiles) of the 
existing bed is adjusted to account for the added sediment mass. 
Previous models use the assumed constant bed density value in each layer 
to solve explicitly for the bed thickness. 

The erosion algorithm simulates the redispersion of stationary 
suspensions by instantly redispersing the thickness of the bed above the 



254 

level at which the bed shear stress, Tj^, is equal to the bed shear 
strength, x^. The average resuspension rate of the partially 
consolidated bed layers over one time-step At is given by an empirical 
law (Eq. 3.13) that is analogous to the rate expression which results 
from a heuristic interpretation of the rate process theory of chemical 
reactions. This rate expression indicates that the resuspension rate 
varies exponentially with the excess bed shear stress. The average 
resuspension rate of the settled bed layers is given by an empirically 
determined expression (Eq. 3.12) that is equal to the first term of a 
Taylor series expansion of the empirical resuspension rate law for 
partially consolidated bed layers. Thus, the rate of erosion of settled 
beds is linearly proportional to the excess shear. The thickness of the 
partially consolidated bed section or the settled bed section eroded per 
time-step is determined using an iteration routine. The effect of 
salinity on the bed shear strength, and hence on the erosion rate of 
that bed, as determined from laboratory resuspension tests is 
incorporated into the erosion algorithm. For a natural mud, the bed 
shear strength was found to double in value, in a linear manner, between 
S = 0 and 2 ppt, and thereafter (for S > 2 ppt) was found to remain 
practically constant. Based on an interpretation of typically observed 
Eulerian concentration-time records in estuaries, erosion is simulated 
to occur only during temporally accelerating flows when t,^ is greater 
than T^. Previous models: 1) assume that the erosion rate of both 
partially consolidated and settled beds varies linearly with the excess 
bed shear stress, 2) do not account for the effect of salinity on the 
bed shear strength, 3) assume that erosion occurs whenever, in either a 
temporally decelerating or accelerating flow, is greater than and 



265 

4) do not account for the bed shear strength and density profile 1n each 
layer (since constant values for and p are used) in determining the 
mass of sediment eroded. 

The dispersion algorithm developed in this study utilizes the 
Reynold's analogy between mass and momentum transfer and solves for the 
four components of the two-dimensional sediment dispersivity tensor 
using the formulation derived by Fischer (1978) for bounded shear 
flows. Thus, only shear flow dispersion is accounted for in this 
algorithm. The limitations of such a dispersion algorithm are 
consistent with those associated with a two-dimensional, depth-averaged 
cohesive sediment transport model. Previous cohesive sediment transport 
models: 1) do not include the cross product dispersion coefficients in 
the advection-dispersion equation and 2) do not include a dispersion 
algorithm to calculate the dispersion coefficients as functions of the 
local depth of flow and the depth-averaged velocity components. 

The settling velocity of cohesive sediments is a function of, among 
other parameters, the suspension concentration, C, the salinity, S, and 
■z^. For concentrations less than C^^ = 0.1 - 0.7 g/1 the sediment 
particles settle independently without much mutual interference, and 
therefore the settling velocity is independent of C. In the range < 
C < 10 - 15 g/1, the settling rate is proportional to C" with n > 

0, due to mutual interference. In the range C > the settling 
velocity decreases with increasing concentration due to hindered 
settling. 

The deposition algorithm integrates the concepts proposed by 
various investigators and represents a unified model of this process. 
Deposition is predicted to occur only in decelerating flows, i.e. 



266 

tjj(t+At) < "^bCt), when is less than the maximum shear stress at which 

deposition can occur, \ . For \ < \ r Change I), where - is the 

'-'max u u,^ 

value of '^^ at which the deposition rate in Range I is equal to that in 

Range IIB (defined below), and for C < Ci for all values of \ < \ , 

"max 

the rate of deposition is determined using the exponential law given by 

Eq. 3.28. For \ r ^ \ ^ \ ^nd C > C-, (Range IIB), the deposition 
"»'- "max ^ 

rate is given by a log-normal expression (Eq. 3.45). The thickness of 
the bed formed by a given deposited sediment mass is determined using 
the properties of the unconsolidated and partially consolidated bed 
sections. As deposition continues, first the unconsolidated layers are 
filled up, followed by the partially consolidated layers. 

Increasing the salinity of the suspending fluid was found to 
increase slightly the settling velocities, and hence the deposition 
rates of a natural mud. The settling velocity, W^, was found to 
increase as s'^*^-^ in Range I. The effect of salinity on the deposition 
rate in Range IIB is given by Eqs. 3.52 and 3.54, and was approximately 
the same as that in Ranges I and IIA. 

The settling/deposition algorithm in previous models: 1) do not 

include the effect of salinity on the rates of deposition and 2) predict 

that deposition occurs only when t^k ^ "^h » either temporally 

min 

accelerating or decelerating flows. Thus, deposition is predicted to 
occur in the previous models during only a small percentage (e.g. 20% 
for kaolinite in tap water) of the shear stress range in which 
deposition has been observed to occur in laboratory steady flow 
experiments. 

The consolidation algorithm accounts for the consolidation of a 
stationary suspension and partially consolidated bed by increasing the 



267 

bed density and bed shear strength with time. Consolidation is 
considered to begin after the bed formation process is complete, at 
which time the bed thickness will be maximum. After two hours of 
consolidation the stationary suspension layer(s) become part of the 
partially consolidated bed, and therefore would undergo resuspension if 
subjected to an excess shear stress. The variation of the mean bed 
density with consolidation time is given by Eq. 3.67, while the density 
profile p{zb) is determined using power law relationships between p and 
(Eqs. 3.70-3.72). The thickness of the bed is reduced to account for 
the expulsion of pore water during consolidation, and to insure that the 
mass of sediment in the bed is conserved. 

The new deposit bed section of the layered bed model is further 
divided into a finite number of strata in order to account for repeated 
periods of deposition, as typically occur in estuaries due to the 
oscillating tidal flow. The top stratum may be composed of a stationary 
suspension and partially consolidated bed, whereas the buried strata are 
composed of just partially consolidated sections. The degree of 
consolidation of a particular stratum is accounted for by using a 
separate consolidation time for each stratum. 

Due to the extremely limited information on bed shear strength 
profiles in cohesive sediment beds, the variation of ^(zk) with 
consolidation time is determine using a power relationship between p and 
-c^, as given by Eq. 3.61. 

None of the previous cohesive sediment transport models account for 
the increase in the bed shear strength and density profiles due to 
consolidation, and thus are not capable of simulating the decrease in 
the susceptibility to erosion of a consolidating bed with time due to 
the continual increase in the bed shear strength. 



268 

The Galerkin weighted residual method is used to solve the 
advection-dispersion equation for the nodal suspended sediment 
concentrations. An existing finite element formulation was modified to 

include the two cross product dispersion coefficients. and D . The 

Aj y X 

model yields stable and converging solutions. The accuracy of the 
solution is affected when the Peclet number becomes too large (greater 
than lo2) or too small (less than 10"^). 

Verification of CSTM-H was carried out against four erosion- 
deposition experiments, three of which were performed in an 18.1 m long 
recirculating flume and the fourth in an 0.76 m mean radius rotating 
annular flume. Simulation of a deposition experiment in an 100 m long 
non-recirculating flume at the Waterways Experiment Station, Vicksburg, 
Mississippi yielded satisfactory agreement between the measured and 
predicted deposit thickness in the downstream 65 m reach of the flume. 
The recorded differences in the upstream 35 m reach may have been caused 
by the occurrence of longitudinal sorting, or the increased turbulence 
present in this section of the flume. CSTM-H is not capable of 
simulating longitudinal sorting because the parameters which 
characterize the log-normal depositional law, used for deposition Range 
IIB, are assumed to be spatially invariant. 

The model was applied to prototype conditions. Sedimentation in 
Camachee Cove Marina, located adjacent to the Intercoastal Waterway in 
St. Augustine. Florida was modeled. A mean shoaling rate of 8.6 
cm/year, which is representative of fair weather conditions only, was 
predicted. Sediment transport in a 10 km hypothetical canal, in which 
both erosion and deposition occurred, was also simulated at four 
different salinities to show the effect of salinity on the rate of 



269 

sediment transport under typical prototype conditions. These three 
simulations demonstrated the significant influence of salinity on the 
rate of erosion of a cohesive sediment bed for salinities less than 10 
ppt, and the reduced effect for salinities greater than about 10 ppt. 

6.2. Recommendations for Future Research 

Based on the conclusions from the present study, the following 
objectives and recommendations for further research are made: 

1. Conduct a thorough laboratory investigation of the 
consolidation characteristics of cohesive soils. Objectives for such a 
research program could be the following: 1) verify and/or modify the 
multi -strata bed formation-consolidation algorithm developed during this 
study. 2) Determine the validity and practicality of using one of the 
existing explicit finite difference finite strain consolidation models 
to predict the consolidation of cohesive soils. 3) Investigate further 
the possible correlation between the density and shear strength of 
cohesive soils. 

2. A laboratory investigation of the redispersion characteristics 
of stationary suspensions and partially consolidated beds is 
recommended. This recommendation is motivated by the differences 
obtained between the measured and predicted suspension concentrations 
during the first one-half hour of periods of erosion in the three 
experiments conducted in the 18.1 m recirculating flume. Erosion 
experiments using flow deposited cohesive sediment beds should be 
conducted in which the consolidation time and applied bed shear stress 
are systematically varied. 



270 

3. The results from the model simulation of the tidal cycle 
experiment in the rotating annular flume reveal the need for studying 
the variation of the vertical suspended sediment profile in an unsteady 
flow field. It would be advantageous to conduct such an experiment in a 
rotating annular flume because longitudinal velocity and suspended 
sediment gradients do not occur in this facility. 

4. The deposition algorithm developed during the present study can 
be modified to include the effect of longitudinal sorting on the rates 
of deposition by incorporating the variation of tgg and cr^ in the two 
horizontal dimensions. Extensive field and laboratory tests would be 
required to determine the following relationships: t^Q = tgg (x.y.x^) 
and 0^ = {x,y,x^). 



APPENDIX A 

DERIVATION OF ADVECTION-DISPERSION EQUATION 



In a diffusing mixture such as the sediment-water binary system, 
the various constituents move at different velocities. For example, the 
vertical advective velocity of the water differs from that of the 
negatively buoyant sediment particles by the sediment setting velocity, 
Wg. The local mass averaged hydrodynamic velocity for a binary system 
is defined as that which would be measured by a pi tot tube: 



^ total momentum C V.. + C.V, 

(A.l) 



Y - WW s s 



mass of mixture C + C 

w s 



Where = velocity of water mass, V = velocity of sediment mass, C 

3 ' w 

mass of water/total volume of mixture and = mass of sediment/total 
volume of mixture. Using the coordinate system defined in Fig. 3.1, Y 
and are defined as 



\ = ui + vj + wk (A. 2) 

= ui + vj + {w+W^)k (A. 3) 

where u, v and w are the fluid velocity components in the respective 
Cartesian coordinate directions. Here the water and sediment particles 
are assumed to be advected in the x- and y- directions at the same 
respective velocity components. 

271 



272 

The advective mass flux of sediment is defined as 



- %\ (A.4) 



while the diffusive mass flux is given by 



J„ = C (V -V) 



s - S^'s"^' (A. 5) 

The diffusive flux in Eq. A. 5 is that due to molecular diffusion, 
which by Pick's first law is equal to 



's- V(-) (A.6) 

P 

where p = density of binary system and d^^ = molecular diffusivity of 
the sediment particles in water. In general D^^^ is a function of Cj, 
C^, ^, T and p, with the latter three parameters being the absolute 
viscosity, temperature and pressure of the binary system. The total 
flux of sediment, N3, is equal to the sum of the advective and diffusive 
fluxes, given by Eqs. A.4 and A.6, respectively. 



h ~- ^s = ^s^ - PDsw'(-) (A.7) 

P 



The law of conservation of sediment leads to the following 
continuity equation for suspended sediment: 



ac 

-^+^.H =S (A.8) 

at ^ 



273 

where S = source/sink term to account for the mass of sediment addded or 
removed per unit volume per unit time. Substituting Eq. A. 7 into Eq. 
A. 8 gives 



ac * C 

~^''^s\= ^•P'^sw'^-^ ^ S (A.9) 
ot p 



In tensor notion Eq. A-9 becomes 

ac 9 a 5 c 

at ax. ^ ^- dx. ^^ax. p 

' J 

In turbulent flow the instantaneous velocity components and 
suspended sediment concentration can be expressed as the 
sum of a time-averaged term and a fluctuating component, i.e. 



V, = + v' 
s. s^ s. 



" ^s S (A. 11) 



where the overbar and the superscript prime denote the mean and 
fluctuating quantities, respectively. The mean term is averaged over a 
time interval, Tj, which is small compared with the time scale for the 
mean flow, but large compared with the time scale for the turbulent 
fluctuations so that the time averages of v' and c' over Tt are 

^ -J 5 1 

approximately zero (MacArthur, 1979). Substituting Eq. A. 11 into Eq. 
A. 10 and averaging the entire equation over Tt gives 



ac a - - a , . a d c 

— + (C V ) + (C Y ) = [pD (-)] + S (A. 12) 

St ax. ^- ax 5x^ swg _ p 



274 

where the subscript s on the sediment concentration C3 has been dropped 
for convenience, and where the following definitions have been used: 

- 1 '''l 1 ^^"^I . 

J C dt c = — / C dt = 0 

(A. 13) 



Tj t Tj t 



1 t+T T t +T 

V = — / V dt V = — / V dt = 0 
i Tj t ^- Tj t ^ 



The terms C V^. in Eq. A. 12 represent the turbulent diffusive mass 
transport of sediment due to the turbulent velocity fluctuations in the 
x^. direction. Reynolds analogy which is based upon the analogy between 
the transfer of mass and momentum in turbulent flow and upon 
Boussinesq's eddy viscosity hypothesis is used to relate these diffusive 
sediment transport terms to the spatial gradient of the time-averaged 
concentration as follows: 



I . 5C 

C V = -E /, ,.v 

s. X. ^ (A. 14 

1 1 Qx. 

where E^. = turbulent diffusion coefficients of sediment in the x- 
direction. Substituting Eq. A. 14 into Eq. A. 12 gives 

ac a a a c a 9r 

Z^Z'^'^^'^i^ ^r-^P'^swr-^-)^ ^— (E, — ) + S (A. 15) 

°t ex. 1 dx ax. p ax. ^i Sx. 

where the bars over the mean parameters have been dropped for 
convenience. 



275 

Next, the fraction of sediment mass, C/p, 1n the molecular 
diffusion term in Eq. A. 15 is simplified by effectively assuming that 
the mixture density or total mass concentration is not a function of 
position in the binary system. The physical justification for this 
assumption is that the effect of spatial gradients of p is included in 
the molecular diffusivity, D^^. As a result, Eq. A. 15 may be rewritten 
as 

9C a 8 5C 

— + — (CV ) = (E ) + S (A. 16) 

at ax. ^• dx. ^ax^. , 

where E^-j = D^^ + E^^^ is the turbulent diffusion tensor, in which the 
off-diagonal terms of . are not neglected. 

Equation A. 16 is the three-dimensional form of the advection- 
dispersion equation for suspended sediment transport in a fully 
developed turbulent flow field. Upon expansion of the tensor terms. Eq. 
A. 16 becomes 



ac a a 5 a ac 

— + — (Cu) + — (Cv) + — {C(w+W )} = — (E — + 

at ax dy 92 ^ ax ^^ax 



ac ac a 5c ac sc 



a ac ac ac 

— (E_ — + E^ — + E — ) + S 

az ^^ax ^^ay ^^az 

The desired two-dimensional form of Eq. A. 17 is obtained by 
integrating this equation over the local flow depth d, which is defined 
as 



276 

d = h(x,y,t) - b(x,y,t) 



(A. 18) 



where h(x,y,t) and b(x,y,t) = elevations of the water surface and bed, 
respectively, with respect to a tidal datum. When vertical integration 
from b(x,y,t) to h(x,y,t) is performed, vertical profiles of the time- 
averaged velocity components in the x and y directions and the suspended 
sediment concentration give rise to dispersion terms. In order to 
account for such terms, the velocity components, u, v and w, and the 
concentration, C, are expressed as the sum of a depth-averaged term and 
a term which is the deviation of the parameter over the depth of flow 
about the depth-averaged value, e.g. 

~ II 

V = V + V 

(A. 19) 

C = C + c" 

where the double bar and the double prime denote the depth-averaged and 
the deviating quantities, respectively. The following definitions of 
the depth-averaged and deviating terms are used: 

e = - J e dz with / e dz = o (a. 20) 

d b b 

for 9 = u,v,w, and C. 

Equation A. 19 is substituted into Eq. A. 17 and the entire equation 
is integrated from b{x,y,t) to h(x,y,t) using Leibnitz rule. The result 
is given on a term-by-term basis below: 



u + u 



w = w + w 



277 



h ac h a = „ 

/ _dz = / — (C+c )dz 

b at b at 
ab 

c"(b,t)— 

at 



a = = a(h-b) 

— {C(h-b)} - c 

at at 



ah 

c (h,t) — + 

at 



(A. 21) 



ha h a = „ = „ a 
J — (Cu)dz = / — {(C+C ){u+u )}dz = — {Cu(h-b)} - 

b ax b ax 5x 

..= ah = 5h ., 5h 

{Cu}| _- {c ull — - {cu }| — - {c u }| — + 

h ax h ax h ax h ax 

a h „ „ 

— / C u dz 

ax b 



(A. 22) 



ha a == == 5h ,.= dh 

J — (Cv)dz = — {Cv(h-b)} - {Cv}l {C v}| 

b ay dy h ay h ay 

ah „ „ 5h a h „ „ 

{Cv }| {c V }| — + — / c V dz (A. 23) 

h ay h ay dy b 



ha =.. = .. 

J — (Cw)dz = {(c+c )(w-H^ )}| (A 24) 

b az h 



h a 

/ — (CW Jdz = {CW }| - {CW }! 
b az 5 2 h ^ b 

ha ac a ac ac a 

b ax ^^dx dx ^^ax ^^Bx ax 

>■ II 

ac ah ac ab a „ ah 

{E ■ }| _+{E }| {E [c] — - 

''''ax h ax ''''ax ^ ax ax ^ ax 

„ ab 

t^c \ — )} (A. 25) 

ax 



278 



J — (Evv— = — {(h-b)E, — } - (E, — )— (h-b) - 



b ay y^dy 



ay 



yy 



ay 



yy 



ay ay 



ac ah ac ab a „ ah 

y^ay h ay yyay b ay ay yy ^ Sy 



„ ab 
ay 



(A. 26) 



h a ac ac 5c 

/ — )dz = {E^^ }| - {E }| 

b az az ^ az b 



(A. 27) 



h a dc a ac ac a 

/ --{E — )dz =_{(h-b)E — } - (E — )— (h-b) - 

b ax ^yay ax ^yay 



ay ax 



ac 



ah 



ac ab a 



ah 



ay h ay 



xy 



ay b ax ax 



xy 



'ay 



ab 
ay 



(A. 28) 



h a ac a ac ac a 

J ^(Eyy— = — {(h-b)E^— } - (E — )— (h-b) 



b ay ^yax 



ay 



xy 



ax 



xy 



ax ay 



ac ah ac ab „ ah 

^yax h ay ^yax bay ^y "ax 

„ Bb 
Cc 1— )} 
''ax 



(A. 29) 



ha ac a h ac 
/ — (E _)dz = — 



V ax ^^az 



/ E 



ax b ^^az 



ac 

- {E }| + 

XZa u 

az h 



279 



9c 

(E }l 

OZ b 



(A. 30) 



ha ac a h ac 
/ — (E^ ~)dz = — / E — 
b Sy y^az dy b y^az 
II 

ac 

y^az b 



ac 

- {E — }| + 
y^az h 



(A. 31) 



h a 5c a = „ 5 = 

/ — )dz = — (C+c )]| - {E — (C+c")}i 

h az '^^ax ^^ax h ^^ax b 



(A. 32) 



ha dc 



5 = 



5 = 



ay 



(A. 33) 



Substituting Eqs. A. 21 through A. 33 into Eq. A. 17 and dropping the 
double overbars from the depth-averaged terms gives 



a a a ahnu ah 

— (dC) + — (dCu) + — (dCv) + — / u"c" dz + — J v"c"dz= 

dy ax b dy b 

a ac ac a 6c ac 

■^^^yy + } + {dE — + dE — } + 

ax ^^ax >^yay ay ^^ax y^ay 

" " 

a h ac a h ac 

— / dz + — / E dz + 

ax b ^^az 8y b y^ay 

h 

H + B + / Sdz (A. 34) 

b 



280 

where H represents the sum of the outward normal flux of sediment and 
change in storage at h(x,y,t), and B represents the sum of the normal 
sediment flux out and storage change at b(x,y,t). The expressions for H 
and B are given below: 



[C+c ] — + [(c+c ){u+u )!—+ [(C+c )(v+v )]^— 
"at %x ■'dy 



II II 



ah 



[(c+c )(w+w )\ - C(C+c )W 1. - E [—(C+c )]^ 

s n XX 



ax 



'ax 



ah 



ac 



ah 



Eyy[— (C+c )],— + [E_ ], - E_[(C+c )],— - 



ay 



ay ^^sz ^ y^' 



'ax 



ah 



ac 



ac 



E^y[— (C+c )\~- - [E.,_ ] 



ax 



■ay 



^ 11 a „ a „ ah 

[E — (C+c )] + [E — (C+c )] - — {E„Jc + 

X^5x " y^ay 



ax 



I, ah 



,1 ah 

ay 



,1 ah 

ax' 



(A. 35) 



II ab „ a „ 5b 

B = -[C^-C [(Cc )W^], . E^/-(C*c )]^-. 

ax ax 



'at 



a II ab ac a „ ab 

EyyMc+c \—- [E — \ + E [— (C+c )].— + 
yy ?^^. D^.. ^ y^ ay l^ax 



ay °ay 



a „ ab ac ac 



ax 



^ay " ^"^^^az ' ^^y^az'b 



° .1 a 5 „ ab 

[E.,— (C+c )] - [E —(C+c )] + — {E [c ]— + 



xz 



ax 



ay 



^ ax ^x 



281 

I. 9 „ Sb 9b 



The terms H and B represent the boundary conditions at h(x,y,t) and 
b{x,y,t), respectively, since Eq. A. 17 was vertically integrated from h 
to b. H is equal to zero since it is assumed that there is no net rate 
of transport of sediment across the instantaneous free water surface. 
Therefore, Eq. A. 34 can be simplified to yield 



0 0 9 11 ■ II 9 

— (dC) + u — (dc) + V — (dC) + — (du"c") + — {dv"c") = 

at 9x 9y dx 9y 

a 9c 9c 9 ac ac 

~~^'^^yx~ ^^vv— ^ + — ^dE ~ + dE — } + 

ax ^^9x ^y9y 9y y>^ax Qy 

» II 

a h 9c 9 h ac h 

Z I ^z7-^' r / ^z7-^^ ^ ^ ^ Sdz (A. 37) 

9x b 9z 9y b b 



where the double overbar denotes the depth-averaged value of the 
quantity thereunder, and where the assumption of an incompressible fluid 
(i.e. V.u = 0) has been utilized. 

A Reynolds analogy is again used to relate the quantities under the 
double overbars in Eq. A. 37 to the spatial gradient of the depth- 
averaged suspension concentration as follows: 



II II 9c 
u.c = -K. . — 

1 IJ; 



'ax, 



1 .J 



1,2 



(A. 38) 



282 

where K^-j = two-dimensional sediment dispersion tensor. Substituting 
Eq. A. 38 into Eq. A. 37 gives 



a 5 a a 5c 5c 

— (dC) + u— (dC) + V— (dC) = — {dD — + dD — } + 
at Qx 5y dx ^^dx ^^dy 



a ac ac a h ac 

— {dD — + dD — } + — / E^^ dz + 

ay y^ax y^ay dx b ^^^z 



II 



a h ac h 

— / E dz + B + / Sdz (A. 39) 

ay b y^az b 



where D^-j = K^.j + Z. ■ = effective sediment dispersion tensor. 

If Ey2 and Ey^ are both assumed to be linear functions of z such 
that the partial derivatives of E^^ and Ey^ with respect to z are 
functions of only x and y, respectively, the two integrals in Eq. A. 39 
become 



a h ac a 

ax b ^^^z ax X2 h xz b 



(A. 40) 



s h ac a 



(A. 41) 



The first and second terms on the right hand sides of Eqs. A. 40 and 
A. 41 should be incorporated into H and B, respectively, as they 
represent fluxes of sediment out of the water surf act and bottom. 



283 

Therefore, Eq. A. 39 becomes 

9 5 a 5 5C dC 

— (dC) + u— (dC) + V— (dC) = — {dD — + dD — } + 

at 9x 5y ax ^^ax ^^ay 

B ac ac 

where S = depth-averaged source/sink term. 

The term B in Eq. A. 42 represents the rate of erosion and 
deposition at the bed-fluid interface, b(x,y,t). That is, sediment that 
settles or diffuses out of suspension is part of the depositional flux, 
and sediment that advects or diffuses into suspension is part of the 
erosional flux. Therefore, B can be alternatively expressed as 



d d 

B = e + Ti = — (dC)| +— (dC)| (A. 43) 

dt Erosive dt Depositional 

Flux Flux 



in which e = idC)] _ = dry mass of sediment eroded per unit 

dt Erosive Flux 
bed surface area, and ti = MdC) |^ ^^^^ ^ ^^^^^ sediment 

dt 

deposited per unit time per unit bed surface area. 

The depth-averaged source/sink term S represents the rate of 

sediment influx to or outflux from the water body not attributable to 

either the processes of erosion or deposition. For example, S would 

account for the removal (sink) of a certain mass of sediment by dredging 

in one area (e.g. navigational channel) of a water body, and the dumping 

(source) of the sediment as dredge spoil in another location in the same 
body of water. 



284 

So Eq. A. 42 is the two-dimensional, depth-averaged advection- 
dispersion equation which governs the transport, addition (i.e. source) 
and removal (i.e. sink) of suspended sediment in a turbulent flow 
field. Expressions for £ and ti are given in Sections 3.4 and 3.6, 
respectively. 

The total dispersive fluxes, f, in the x- and y- directions are 
given by 

ac ac 

f„ = dD — + dD — (A. 44) 

X xxg^ xy^y ^'^•'^^^ 

ac ac 

f = dD — + dD — (A 4"^^ 

y yxa^ yy^^ ^'^•^s) 

which are the expressions inside the curly brackets in Eq. A. 42. Using 

the coordinate system shown in Fig. 3.1, the dispersive fluxes in the 

longitudinal (along the flow axis) and transverse (perpendicular to flow 

axis) directions, f-, and f^, respectively, are related to f, and f„ as 

' X y 

follows: 



-f^cos Q - f^sin e 



fy = f^sin 9 + f^cos e 



(A. 46) 



(A. 47) 



where f-, and f^ are given by 



ac 

f, = -dD, — (A. 48) 

ac 

f^ = -dD — (A. 49) 

an 



285 

in which © = arctan (u/v), D-j = effective longitudinal dispersion 
coefficient and = effective transverse dispersion coefficient. The 
four components of the two-dimensional dispersion tensor, D^-j, are 
related to D-] and by the following functional relationships: 

D^^ = D^cos^e + D^sin^e (A. 50) 

'^xy = V =V^D^-D^)sin(2e) (A. 51) 

D = D^sin^e + D^cos^e (A. 52) 



Expressions for D^-j are given in Chapter III, Section 3.5. 



APPENDIX B 

COEFFICIENT MATRICES IN THE ELEMENT MATRIX 
DIFFERENTIAL EQUATION 

The purpose of this appendix is to list the matrices and vectors in 
the element matrix differential equation (4.23) and describe how they 
and the contour integral in Eq. 4.22 are evaluated. 

The product of the element coefficient matrix, [k], and the nodal 
concentration vector, {C}^, is seen from Eqs. 4.22 and 4.23 to be equal 
to 

A 

.dc .5c ^'U ac 

[k]{c}^ = // [N (u— + v— ) + [D — + 

e ax ay 5x ^^ax 



ac ac^ dc 



where the approximate velocities, u and v, are evaluated at each point 
(C,Ti) as follows: 

l=n ^ i=n 

u = 2 N.u. v = 2 N.v. (B.2) 

1=1 ^ ^ 1=1 ^ ^ 

The dispersion coefficients are considered not to vary significantly in 
either space or time in this formulation, and therefore are assumed to 
be constants. The transformation from global coordinates to local 

element coordinates derived in Chapter IV, Section 4.4.2, gives 

286 



287 

dxdy = lJ|5dTi 



{B.3) 



Substituting Eqs. B.2 and B.3 into Eq. B.l and dividing through by the 
concentration vector {C}^ gives the following expression for the (i,j) 
term of [k]: 

' ^ ^ ax ^ ^ay ax ax 

aN. an . aw. aN . an. aN. 

a7 ~ ^y'^ a7 aJT "^yy a7 

|J|dCdTi for 1=1, n (B.4) 



The notation 1=1, n indicates that the variables with the subscript 1 are 
summed from 1=1 to l=n=number of nodes in a particular element. Thus, 
the element coefficient matrix is given by 

i=n j=n l=n 

[k] = 2 Z E k (B.5) 
i=l j=l 1=1 

The double integration in Eq. B.4 is performed using the Gauss- 
Legendre quadrature formula as follows (Ariathurai et a1_., 1977): 

. , m=NQ mm=NQ 

/-I i_i f(^,il) d^dTi = S 2 H H f(| ,Ti ) (B.6) 
^ ^ m=l mm=l ™ 



288 

where f=k Is given by Eq. B.4, NQ= number of quadrature points and 
H=weight factors. 

The element temporal matrix [t]^ is given by 

i=n j=n l=n 

Itl = >: z z // N H N d, [J! d^dn (b.7) 
i=l j=l 1=1 % 1 J ' I 

where again the integration is performed using the Gauss-Legendre 
method. 

e 

The element source/sink vector {f} is given by 



i=n j=n l=n 



if} = - ^ Z Z // N.NTd,S|J|dldn (B.8) 
i=l j=l 1=1 e ^ ' ' 

and the boundary matrix [b]^ is given by 



i=n j=n l=n ' j- 

2 2 E / M N d, [(D + D^^ ) n^ 

i=l j=l 1=1 ^ 1 ' XX 5x ay X 



The boundary matrix [b]e accounts for a specified concentration flux 
boundary condition along the boundary of a domain boundary element. 



APPENDIX C 
COMPUTER PROGRAM 



The computer program of CSTM-H is written in FORTRAN IV using 
double precision arithmetic. Double precision is required in simulating 
laboratory scale tests in order to minimize round-off error. Prototype 
systems can be modeled using single precision arithmetic. A description 
of the main program and the subroutines followed by a flow chart and 
user's manual are presented. 

C.l. Main Program 

In the main program the following information is read: 
input/output file numbers, problem option and control parameters, finite 
element grid geometry and transient control parameters. The I/O file 
numbers determine the file numbers used in both reading and printing 
certain data. The problem option parameter, NOPT, specifies which one 
of four types of problem is to be solved. The four types of problems 
that can be solved by CSTM-H are: 1) steady state advection-dispersion 
of a conservative constituent, 2) unsteady advection-dispersion of a 
conservative constituent, 3) cohesive sediment transport and 4) one- 
dimensional consolidation of a clayey sediment bed. The flow chart in 
Section C.3 diagrams the program steps used in the solution of each 
problem type. The transient input code arrays specify at which time 

steps new boundary conditions, flow depths, velocity field, dispersion 

289 



290 

coefficients, sediment settling velocities, density (i.e. salinity) 
field or local source/sink terms (explained in Appendix A) are read in 
or calculated using either built-in algorithms or user supplied 
routines. The user's manual in Section C.4 gives a short definition of 
each parameter read in either the main program or one of the 
subroutines. The grid data that need to be read includes the number of 
nodes, NP, number of elements, NE, nodes forming each element (nodal 
connections in counterclockwise direction) and the two horizontal 
coordinates of each node. 

The main program also initializes necessary arrays, prints out the 
initial conditions and initial values of certain parameters, contains 
the main time loop and prints out the concentration-time record for 
specified elements. 

C.2. Subroutines 

A brief description of the subroutines and subf unctions in CSTM-H 
is given below. 

Subroutine BAND - Forms and solves the system matrix equation at each 
time step for the nodal concentrations using Gaussian elimination. This 
subroutine is used when ISOLV = 0. 

Subroutine BEDFOR - Forms the sediment bed that is a result of 1) 
deposition during the previous time step or 2) new deposits present on 
top of the settled bed at the start of the modeling. 
Subroutine BEDMOD - Control program for bed formation and consolidation 
routines. Determines mass of stationary suspension for which T^^^, = 2 
hours; calls Subroutine BEDFOR, and stationary suspension becomes part 
of the partially consolidated bed. Determines the consolidation time 



291 

for each bed stratum, and calls Subroutine CONSOL for the strata with 
Tjj, > 2 hours. 

Subroutine BEDSS - Computes the bed shear stress at each node. 
Calculates the average flow depth, velocity, bed shear stress, water 
density and concentration for each element using the element shape 
functions. Compares this time step's bed shear with the previous time 
step's bed shear to determine if the flow at each node is temporally 
steady, accelerating or decelerating. 

Subroutine COMPAR - Compares analytical solution with numerical solution 
for steady state problems. 

Subroutine CONCBC - Reads or computes concentration boundary conditions 
at specified nodes. 

Subroutine CONCIC - Reads or computes the initial suspended sediment 
(for sediment transport problem) concentration at every node. 
Subroutine CONSOL - Computes the increase in the bed density profile in 
each stratum due to consolidation. The bed strength profile is computed 
as a function of the new bed density profile. 

Subroutine DENSITY - Computes the water density at every node using the 
given water temperature, salinity and suspension concentration. The 
kinematic viscosity is also calculated as a function of the water 
temperature. 

Subroutine DEPMAS - Computes the dry sediment mass deposited during the 
previous time step for every element where deposition is predicted to 
occur. 

Subroutine DEPSN - Computes the rate of deposition at each node where 
deposition is predicted to occur. 



292 

Subroutine DEPTH - Reads or computes (using user specified procedure) 
the depth of flow for each node at the time steps where the appropriate 
transient code array indicates that a change in depth occurs. 
Subroutine DISPER - Reads or computes (using the dispersion algorithm) 
the four components of the two-dimensional dispersion tensor for each 
node at every time step where the appropriate transient code array 
indicates that the values of the dispersion coefficients change. 
Subroutine DRYNOD - Determines at which nodes and elements the water 
depth is negative (i.e. dry). These nodes are eliminated from the 
system array coefficient matrices. 

Subroutine ELSTIF - Forms the element coefficient and load matrices. 
Modifies element load matrix to account for specified boundary 
conditions. 

Subroutine EXACT - Computes analytical solution to steady state 
convection-diffusion problem. 

Subroutine FRONT - Forms and solves the system matrix equation at each 
time step for the nodal concentrations using the frontal elimination 
routine. This subroutine is used when ISOLV = 1. 
Subroutine ITERC - Computes thickness of partially consolidated bed 
formed by specified dry mass of sediment using an iteration procedure. 
Called by Subroutine BEDFOR. 

Subroutine ITERM - Computes thickness of stationary suspension formed by 
specified dry mass of sediment using an iteration procedure. Called by 
Subroutine BEDFOR. 

Subroutine LOAD - Forms the array NBC which numbers the equations in the 
system matrix. Number of equations is equal to the number of nodes 
minus the number of nodes at which boundary conditions are specified. 



293 

Computes the band width for the system coefficient matrix. This 
subroutine is called when Subroutine BAND is used to solve the system 
matrix. 

Subroutine LOADX - Forms the array NBC. This subroutine is called when 
Subroutine FRONT is used to solve the system matrix. 
Subroutine ORGBED - Reads the original settled bed profile and the 
initial dry mass per unit bed surface area of new deposits on top of the 
settled bed for elements where such exists. 

Subroutine RECORD - Records the values at each time step of various 
parameters for the elements where time records are desired. 
Subroutine RED - Called by Subroutine FRONT to read data from temporary 
disc data file. 

Subroutine REDISP - Computes the redispersion rate for unconsolidated 
new deposit layers (stationary suspension) when the flow is accelerating 
and the bed shear stress is greater than the shear strength of the 
suspension surface. Computations are made on an element-by-element 
basi s. 

Subroutine RESUSP - Computes the resuspension rate for exposed partially 
consolidated bed layers or settled bed layers when the flow is 
accelerating and the bed shear stress is greater than the shear strength 
of the bed surface. Computations are made on an element-by-element 
basi s. 

Subroutine SEDPRP - Reads the settling velocity, new deposit and 
consolidation properties of the cohesive sediment. Settling velocity 
parameters and the new deposit properties are printed out. 
Subroutine SETVEL - Reads or computes using a built-in algorithm the 
sediment settling velocity in Range I and for C < Ci in Range II as a 



294 

function of the suspension concentration and salinity at each node for 
the time steps where the appropriate transient code array indicates that 
a change in the settling velocity occurs. 

Subroutine SHPFNS - Computes the isoparametric quadratic shape functions 
and their derivatives for quadrilateral elements with parabolic sides. 
Subroutine TSHAPE - Computes the isoparametric quadratic shape functions 
and derivatives for triangular elements with parabolic sides. 
Subroutine VEL - Reads or computes using a user specified routine the 
depth-averaged components of the velocity in the x- and y- directions at 
each node for the time steps where the appropriate transient code array 
indicates that a change in the velocity occurs. 
Subroutine WRITER - Prints out the bed shear stress, bed elevation, 
erosion/deposition rates and the layer-by-layer bed properties for each 
element at each time step where the appropriate transient code array 
specifies. 

Subroutine WRT - Called by Subroutine FRONT to perform mass transfer of 
data to temporary disc data file. 

Function DENFUN - Computes the water density as a function of 
temperature and salinity. 

Function FBETA - Computes the value of the empirical coefficient A in 
Eq. 3.68 as a function of T^^.. 

Function FDELTA - Computes the value of the empirical coefficient B in 
Eq. 3.68 as a function of J^^. 

Function FMBDN - Computes the value of the final mean bed density (given 
by eq. 3.64) as a function of Cq. 

Function FTCIN - Computes the value of T^^^.^^ (given by Eq. 3.65) as a 
function of Cq. 



295 

Function FZHMIN - computes the value of z^^^^ (given by 3.69 with 9 = 
z^^^) as a function of T^^,. 

Function SIGFUN - Computes the value of 02 (given in Eq. 3.36) as a 
function of t;|^. 

Function T50FUN - Computes the value of tgg (given in Eq. 3.36) as a 
function of Tj^. 



C.3. Flow Chart 



296 



START 



J 



READ I/O 
FILE NUMBERS 




COORDINATES 




NO 




297 



0 



1 




/ 

READ TRANSIENT 
INPUT DATA 







READ ELEMENT 
NUMBERS FOR WHICH 
TIME HISTORY IS TO 
BE PRINTED OUT 




INITIALIZE 
NECESSARY 
ARRAYS 



READ AVERAGE 
WATER TEMP. 
AND INITIAL 
SALINITIES 



READ SOURCE/SINK 

TERf-i AT 
APPROPRIATE NODES 





YES 



SEDIMENT PROBLEM - 
INITIALIZE BED 
PROPERTIES 



5 



6 



299 




i 





CALL DENSTY- 




ELEMENTAL BED SHEAR 




STRENGTHS AND BULK 




DENSITIES CALCULATED 




AS A FUNCTION OF ' 




ELEMENTAL SALINITY 




VALUES 




300 



CALL VEL- 
READ INITIAL 
VALUES OF FLOW 
VELOCITIES 




CALL BEDSS- 
CALCULATE 

NODAL 
BED SHEAR 
STRESSES 



CALL DISPER- 
READ/CALCULATE 
INITIAL DISPERSION 
COEFFICIENTS 






CALL SETVEL- 


YES 


READ/CALCULATE 




NODAL SETTLING 




VELOCITIES 




301 



1 




CALL 


LOAD- 


NUMBER OF EQUATIONS 


IN THE SYSTEM COEF. 


MATRIX AND 


BANDWIDTH 


DETERMINED 



CALL DRYNOD- 
DETERMINES WHICH 
NODES AND 
ELEMENTS ARE DRY 



CALL LOADX- 
NUMBER OF EQUATIONS 
IN THE SYSTEM COEF. 
MATRIX AND BANDWIDTH 
DETERMINED 



PRINT INITIAL 
CONDITIONS AND 
SEDIMENT PROPERTIES 



STEADY-STATE 


YES 


PROBLEM 






NO ^ 


UNSTEADY AND 


SEDIMENT 






PROBLEfIS- 



YES 



IF 
ISOLV 
EQ 



NO 



NOPT 
EQ 



NO 



YES 



302 



CALL FRONT- 
NODE SOURCE 
TERMS ADDED, 
GLOBAL COEF. 
MATRIX FORMED 
AND SOLVED 
BY FRONTAL 
ELIMINATION 
ROUTINE USING 
FULL PIVOTING 



YES 



CALL BAND- 
NODE SOURCE TERMS 
ARE ADDED INTO 

SYSTEM LOAD 
MATRIX, GLOBAL 
COEF. MATRIX FORMED 
AND IS SOLVED 
BY GAUSSIAN 
ELIMINATION 




CALL DEPSN- 
DEPOSITION 

RATES 
CALCULATED 



CALL RESUSP- 
SURFACE 
EROSION 
RATES 

CALCULATED 




NO 



CALL 


WRITER - 


PRINT 


CONCS. 



CALL ELSTIF- 
ELEMENT STIFFNESS 
ARRAYS FOR FIRST 
TIME STEP FORMED 



CALL COMPAR- 
COMPARE WITH 
EXACT SOL. 





MAIN TLME LOOP 
DO N=2,iNTTS 




DEPENDING ON INPUT 
CODES, READ NEW 
PARAMETERS FOR 
THIS TIME STEP 




CALL DENSTY- 
SET NEW 
SALINITIES 



CALL CONCBC- 
SET NEW 
BOUNDARY 

CONDITIONS 



CALL DEPTH - 

SET NEW 
FLOW DEPTHS 



YES 





CALL DISPER- 
SE! NEW 
DISPERSION 
COEFFICIENTS 



CALL SETVEL- 
SET NEW 
SETTLING 

VELOCITIES 



READ NEW 
NODAL SOURCE/ 
SINK TERMS 



305 




" CALL 


DRYNOD 






CALL 1 


-OADX 



YES 




CALL DEPSN 
CALL REDISP- 
REDISPERSION 
RATES CALCULATED 
CALL RESUSP 




CALL BAND 



CALL FRONT 




306 



IF 
iNOPT 
EQ 



NO 



YES 



CALL DEPSN- 
DEPOSITION 
RATES CALCULATED 



CALL BEDMOD- 
CONTROLS BED 
CONSOLIDATION 
ALGORITHM 




CALL RECORD- 
OUTPUT FOR THIS 
TIME STEP SAVED 



II 



307 




308 



C.4. User's Manual 



SET A 

CARD A.l 

1-5 
6-10 
11-15 
15-20 
21-25 
26-30 
31-35 
36-40 
41-45 
46-50 
51-55 
56-60 
61-65 



CARD A. 2 
1 

2-78 
CARD A. 3 

1-5 



JOB CONTROL CARDS 



6-10 



11-15 
16-20 



(1315) 

IN 

LP 

INC 

IND 

INE 

INF 

ING 

INH 

INI 

INB 

INS 

INSS 

ISOLV 



(11, 19A4) 
NSTOP 

TITLE 
(1615) 

NOPT 



ICODE 



NTTS 
IVEL 



I/O file numbers and equation solver used 

General input filt number (default 5) 

Output file number (default 6) 

Initial concentrations 

Diffusion coefficients 

Mode point bottom elevations (initial) 

Node point flow velocities 

Settling velocities 

Flow depths 

Finite element grid geometry data 

Boundary conditions 

Salinities 

New nodal . salinities 

0 - uses band solver 

1 - uses frontal solution technique 

Job stop and title 

0 - continue 

1 - end of job 

Job title 

Job control parameters, input codes, and 
problem options 

Type of problem 

1 - steady state transport problem 

2 - unsteady transport problem 

3 - sediment transport problem 

4 - consolidation problem only 

Output control for non-sediment problems 

0 - standard output 

1 - compares with analytic solution 

calculated in Subroutine EXACT 

Number of time steps 

Determines initial velocity field, i.e., 
at time step #1 (for unsteady problems 
only) 

1 - velocity components in x and y 

directions are set equal to 
constants CONXV and CONYV read 
in Subroutine VEL 

2 - each nodal velocity read in from 

input file INF 

3 - velocity computed using user supplied 

routine in Subroutine VEL 



309 



21-25 : lELEV 



26-30 : IDIFl 



31-35 : IBED 



36-40 : ISET 



41-45 : IDEP 



46-50 : ICONC 



51-55 : INBC 



46-60 : IDRY 



Elevation of bottom above a given datum 
at node points 

0 - all elevations set equal to 0.0. 

1 - read each value read in from file 

number INE 

Initial diffusion coefficient values at 
each node 

1 - and D are set equal to constants 

read in Subroutine DISPER 

2 - nodal diffusion coefficients are read 

in from file number I NO 

3 - diffusion coefficients are calculated 

using user supplied procedure 

Initial bed profile 

0 - no sediment present on bed 

1 - bed profile read in Subroutine ORGBED 

Initial settling velocity at each node 

1 - set to a constant read in Subroutine 

SETVEL 

2 - each settling velocity is read in 

from file number ING 

3 - settling velocities are computed from 

model in Subroutine SETVEL 

Initial depts of flow at each node 

1 - set to constant read in Subroutine 

DEPTH 

2 - read in from file number INH 

3 - computed accordign to user supplied 

procedure in Subroutine DEPTHS 

Initial suspended sediment concentrations 

1 - set to constant 

2 - read in from file number INC 

3 - computed according to user supplied 

procedure in Subroutine CONCIC 

Boundary conditions 

1 - each value read in from file number 

1MB 

2 - computed in Subroutine CONCBC using 

user supplied routine 

Code to indicate dry node (i.e. negative 
flow depth) problem 

0 - no dry nodes will occur 

1 - possible dry nodes 



51-65 : ISS 



Code to indicate whether sediment trans- 
port problem occurs in steady or unsteady 
flow 



310 



0 - unsteady flow 

1 - steady state flow 

66-70 : ISOUR Code to indicate if local sediment source 

of sink is located at any node 

0 - no source/sink 

1 - source/sink occurs at one or more 

nodes 



SET B MESH DATA 

These data are read unformatted from file unit INI 
CARD B.l 



NE Number of elements in system 

NP Number of nodes in system 



CARD B.2 et.seq. 



NOPd.K) Nodal connections read 

counterclockwise (8 percard for 
quadrilateral element, 
6 for triangular element) 



CARD B.3 et.seq. 



C0RD{J,1) X-coordinate (meters) 
C0RD(J,2) Y-coordinate (meters) 



SET C 



TRANSIENT PROBLEM INPUT 



CARD C.l 



TETA 



DT 

TIM(l) 
CARD C.2 et.seq. (8011) 
l,etc. : NPMA(I) 

for I=1,.„.,NTTS 



Transient input 

Degree of Implicitness for Crank-Nicholson 
time marching scheme 

0 - explicit 

1 - implicit 

Time step - sees (should be of the order 
600-1800 seconds for sediment transport 
problems in estuaries) 

Starting time - sees 

Code to change time step 

The value of NPMA at each time determines 
if the time step will be changed 

0 - no change 

1 - double time step 

2 - halve time step 



311 



CARD C.3 et.seq.(80Il) 
l,etc. : IFFd.l) 



CARD C.4 et.seq.(80Il) 
l.etc. : IFF{I,2) 



CARD C.5 et.seq.(80Il) 
l.etc„ : IVCOD(I) 



CARD C.6 et.seq.(80Il) 
l.etc. : IDIF(I) 



CARD C.7 et.seq.(80Il) 
l,etc. : IDEPC(I) 



Code for new boundary conditions 

Determines if there are new boundary 
conditions 

0 - no change in boundary conditions 

1 - each value read in from cards 

2 - computed in Subroutine CONCBC using 

user supplied procedure 

3 - each value read in from file number 

INB 

Output control 

0 - no output 

1 - sedimentation data only 

2 - concentrations only 

3 - concentrations and sediment transoort 

data ■ 

New velocities 

Same as IVEL but for each time step 

0 - no new velocities 

1 - X 8 Y velocities set equal to 

constants CONXV and CONYV read in 
Subroutine VEL 

2 - each nodal velocity read in from 

input file INF 

3 - velocity computed using user supplied 

routine in Subroutine VEL 

New dispersion coefficients 
Same as IDIF 

0 - no new dispersion coefficients 

1 - D^ and D are set equal to constants 

read in Subroutine DISPER 

2 - nodal dispersion coefficients are 

read in from file IND 

3 - dispersion coefficients are 

calculated using dispersion 
algorithm 

New depths of flow 

Same as IDEP 

0 - no new depths 

1 - set to constant read in Subroutine 

DEPTH 

2 - read in from file IHN 

3 - computed according to user supplied 

procedure in Subroutine DEPTH 



312 



CARD C.8 et.seq.(80Il) 
l.etc. : ISALC(I) 



CARD C.9 et.seq.(80Il) 
l.etc. : ISVS(I) 



CARD C.IO et.seq.(80Il) 
l.etc. : ISORS(I) 



CARD C.ll 



NHIS 



NELH 



Mew salinities 

0 - no new salinities 

1 - set to a constant read in Subroutine 

DEM STY 

1 - new salinities at specified nodes are 

read in Subroutine DENSTY 
3 - salinities for all nodes are read in 

Subroutine DENSTY 

Only for sediment problems H0PT=3 

New settling velocities. Same as ISET 

0 - no new settling velocities 

1 - set to a constant read in Subroutine 

SETVEL 

2 - each settling velocity is read in 

from file number JNG 

3 - settling velocities are computed from 

model in Subroutine SETVEL 

Code for local sediment source or sink 

Same as ISOUR 

0 - no source/sink 

1 - source/sink occurs at one or more 

nodes 

Output control 

Number of elements for which time 
history will be written 

Element numbers 



For Steady State Sediment Problems (ISS.NE.O), specify at which nodes and 
elements erosion and deposition initially (i.e. first time step) occur 

CARD C.12 et.seq.(80Il) Code for nodes 



l.etc. : ISTP(I) 
for 1=1,..., NP 



0 - deposition occurs initially 

1 - erosion occurs initially 



CARD C.13 et.seq.(80Il) Code for elements 



l.etc. : ISTE(I) 
for I=1,...,NE 



0 - deposition occurs initially 

1 - erosion occurs initially 



SET D WATER AND SEDIMENT PROPERTIES 

Read in Subroutine DENSTY 



313 



CARD D.l 



(FIO.5,110) Water parameters 



1-10 
11-20 



TMP 
IS 



If IS = 0: 

CARD D.2 (F10.5) 

1-10 : SW 
If IS = 1: 

CARD D.2 et.seq.(7F10.5) 
1-10 : SAL(I) 



Average water temperature CO 
Determines how initial salinities are 
read in 

0 - constant salinity for all nodes 

1 - salinity for each node is read in 



Constant salinity 

Value of constant salinity - ppt 

Nodal salinity values 

rth 



Salinity value for lIIL node - ppt 

If ISOUR.NE.O, read source/sink term at appropriate nodes 

CARD D.3 (4(110, FIO. 5) 

1-10 : IT(J) Node number 

11-20 : TEMP(J) Local source/sink term - Kg/m^ 

Reading stops for IT(J) < 0 

Read in Subroutine SEDPRP for sediment problems (N0PT=3) 
CARD D.4 



1-10 
11-20 
21-30 
31-40 

CARD D.5 

1-10 
11-20 
21-30 
31-40 
41-50 
51-60 

CARD D.6 

1-10 
11-20 



(4F10.5) 


Settling velocity 


parameters 


CRCN=C, 


See equations 


for 




below - 


CRCN2=Cp 


See equations 


for 




below - 


CRCN3=C^ 


See equations 


for 




below - 


GAC 


Density of sediment mineral 


(6F10.5) 










AA=Ai 


See equations 


for 




below 


AB=Ao 


See equations 


for 


^s 


below 


AC=Ao 


See equations 


for 


^^s 


below 


B 


See equations 


for 


< 


below 


F 


See equations 


for 


Ws 


below 


AL 


See equations 


for 


^^s 


below 


(2E10.3) 










WSl=W3i 


See equations 


for 




below - 



Equivalent sediment particle diameter at 
^50 ■ 



314 



CARD D.7 



(2F10.5) 



1-10 
11-20 

CARD D.8 

1-10 
11-20 
21-30 

CARD 0.9 

1-10 
11-20 



EXPNl=ni 
EXPN2=n2 

(3F10.5) 

EXPMl=mT 
EXPM2=nip 
EXPM3=m3 

(2E10.3) 



WSK1=K 
WSK2=K; 



1 



See equations for below 
See equations for below 



See equations for W below 
See equations for W- below 
See equations for W below 



See equations for below - m/s 
See equations for below - m/s 



NOTE: For RANGE I and C < C^ in Range II 

= AA*WS1*(SAL)**EXPM1 for C < CRCNl where SAL = salinity 
If (SAL < 0.1 ppt)SAL = 0.1 ppt 

= AA*WSK1*C**EXPN1*(SAL)**EXPM1 for CRCNl < C < CRCN2 

W3 = AB*WSK2*C**EXPN2*(SAL)**EXPM2 for CRCN2 < C < CRCN3 

Wg = AC*G*D**2*(GAC/GAW-1)*250*(C/CRCN3-1)**AL*(SAL)**EXPM3/ 
(18*v*D**1.8) for C > CRCN3 

If there is only one W = kc" relationship between C = CRCNl and 
the concentration at which hindered settling begins set CRCN3 = 
CRCN2 in CARD D.4. 

NOTE: For C > C^ in Range II 

T=AL0G10((T/T5q)*B*(SAL)**F)**(1./SIG2) 
CARD D.IO (2I10, 2F10.5) Properties of new deposits 



I-IO 
11-20 
21-30 
31-40 



NLAYTM 
NLAYT 
TAUMIN 
TAUMAX 



Number of layers formed by unconsolidated 
new deposits (UND) 

Number of layers formed by partially 
consolidated new deposits (CND) 



mm 



max 



Parameters characterizing functional relationship between and log^Q 
(tgQ) and CT^ at a salinity of 35 ppt. 



CARD D.ll 

1-10 
11-20 



(5F10.5) 

Al 
Sl 



See equations for 02 below 
See equations for 02 below 



315 





A2 




See 


equations for 


31-40 : 


S2 




See 


equations for 


41-50 : 


CI 




See 


equations for 


CARD D.12 


(5F10.5) 






i-in 


A3 




See 


equations for 


11-20 : 


S3 




See 


equations for 


21-30 : 


A4 




See 


equations for 


31-40 : 


S4 




See 


equations for 


41-50 


C2 




See 


equations tor 


NOTE: For 




CI : 


a, = 


Sl*-ug + Al 




t > 


CI : 


^2 = 


S2*i;g + A2 






C2 : 


%0 = 


eo.^io^s^*"^ 


For 




60.*10(S3*i:g 






C2 : 


^50 " 



J50 
•^50 
^50 
^50 

So 



below 
below 
below 
below 
below 



CARD D.13 



+ A4) 



et.seq.{4(2F10.5)) Shear strength and dry sediment 
density for unconsolidated new deposit layers. NLAYTM 
pairs of values are read in starting at the bed 
surface and proceeding down to the bottom of the 
bottom UND layer. 



1-10 
11-20 



SSM(I) 
GADM(I) 



Bed shear strength - N/m^ 
Dry sediment density - kg/rn^ 



for I = 1,..., NLAYTM 
CARD D.14 et.seq.(F10.5) 



1-10 

for I 
CARD D.15 



: TLAYM(I) 
1,..., NLAYTM 



Thickness of unconsolidated new deposit 
layer - m 



et.seq. (2F10.5) Shear strength and dry sediment 
density for partially consolidated new deposit layers. 
NLAYT+1 pairs of values are read starting at the top 
of these layers and proceeding downward. 



1-10 
11-20 

for I 

CARD D.16 



1-10 
11-20 
21-30 



SS(I) 
GAD(I) 



Bed shear strength - N/m^ 
Dry sediment density - kg/rrr' 



1,..., NLAYT+1 

et.seq. {3F10. 5) Thickness, and oc values for 
each partially consolidated new deposit layer. 



TLAY(I) Layer thickness - m 

EPSLON(I) - kg/m^/s 

ALFA(I) a - dimensionless 



315 

for I = 1,...,NLAYT 

NOTE: The properties read 1n on CARDS D.13-D.16 are determined from 
laboratory experiments (see Appendix D, Section D.2 for a 
description of these experiments). These are the 
properties assigned to new deposits if/when deposition occurs 
during model simulation or initially if new deposits are 
present on top of the original settled bed, as specified in 
SET K. 



CARD D.17 



1-10 
11-20 
21-30 



CARD D.18 

1-10 
11-20 
21-30 
31-40 
CARD D.19 



1-10 
11-20 

CARD D.20 



1-10 
11-20 



1-10 
11-20 



(3F10.5) Empirical coefficients in expression for 
mean bed density as a function of 
consolidation time, T^ . 

^c 

AP f in Eq. 3.75 

ALAMDA p in Eq. 3.75 

TCC Time at which the coefficients A and B in 

Eq. 3.76 become constants 

(4F10.5) Empirical coefficients in expressions for 
final mean bed density and corresponding 
consolidation time, T^^, . 

FMBDO (pJq in Eq. 3.72 

AI a in Eq. 3.72 

TCINO ^^dcjo 1'" ^<^' 3.73 

BI b in Eq. 3.73 

(2F10.5) Empirical coefficients in expression for 
A (in Eq. 3.76) given by Eq. 3.77. 

AQ D (with 9 = A) in Eq. 3.77 

BQ F (with 0 = A) in Eq. 3.77 

(2F10.5) Empirical coefficients in epression for 
B (in Eq. 3.76) given by Eq. 3.77. 

EQ D (with 9 = B) in Eq. 3.77 

FQ F (with 9=8) in Eq. 3.77 



CARD D.21 (2F10.5) Erppirical coefficients in expression for 



max 



given by Eq. 3.77 



PQ 
SQ 



For NOPT = 2 or 3: 



D (with e = z[^^^) in Eq. 3.77 
^ ^""'^^ ® = ' max) 1" Eq. 3.77 



SET E 



INITIAL CONCENTRATION FIELD 



The initial concentration at each node must be specified for all unsteady 
problems. The type of Input is determined by the value of ICONC. 



317 

Read from file unit INC 

ICONC = 1 Initial concentration set to a constant at all nodes. 
CARD E.l {F10.5) 

1-10 : CINT Initial concentration - kg/n? 

^CONC = 2 Read in initial concentration for each node. 
CARD E.l et.seq.(4(I10,F10.5)) 

1-10 : IT{J) Node number 

11-20 : TEMP{J) Initial concentration - kg/m^ 

Reading stops for IT(J) < 0 

^CONC = 3 Compute initial concentrations at each node using user 
supplied model in Subroutine CONCIC. 

For NOPT = 3 (Sediment Problems) : 

SET F INITIAL BED ELEVATIONS 

If IELEV+0, the initial bed elevation, with respect to some datum, at 
each node is read in. 

Read from file unit INE 

CARD F.l et.seq.(8F10.5) 

1-10 : ELEV(I) Bed elevation for node I - m 

I = 1,...,NP 

For NOPT = 3: 

SET G ORIGINAL SETTLED BED PROFILE 

Read in only if IBED is not zero. Otherwise the default bed condition 
will be a clean bed. 

CARD G.l et.seq.(3I5,F10.5) for each element 

Element number 

Number of layers of original settled 
bed for element NN 

If NM=0, bed properties are read in for 
each element. If NM=i=0, constant values 
are read in and used for all elements. 
Average density of pore water in original 
bed - kg/m3 

CARD G.2 et.seq.(2F10.5) Shear strength and dry sediment 

density for original settled bed layers. NLA+1 



1-5 


: NN 


6-10 


■ NLA 


11-15 


NM 


16-25 : 


GWA 



318 

pairs of values are read in starting at the top layer 
and proceeding downward. The first values are for the 
top of the original bed. 

1-10 : SSTO(NN,L) Bed shear strength - N/m^ 
11-20 : GADO(NN,L) Dry sediment density - kg/m^ 

For I = 1,...,NLA+1 

CARD G.3 et.seq.(2E10.3) Thickness and value of M for each 

settled bed layer. 

1-10 : THICKO(NN,I) Thickness of Ith_ layer - m 
11-20 : EROCON{NN,I) M value for I th layer - kg/m^/s 

For I = 1,...,NLA 

Note: CARDS G.l, G.2 and G.3' are repeated for NM=1,...,NE 

when NM=0. When NM*0, these cards are read in only 
once. 

If stationary suspension is present on top of original settled bed, set 
NN=-10 at the end of the above set (i.e. after CARDS G.l, G.2 and G.3). 
For NN=-10, read the following cards. 

CARD G.4 et.seq.(I10,F10.5) for each element 

1-10 : IT(J) Element number 

11-20 : TEMP(J) Dry mass per unit area of (soft 

unconsolidated sediment) on top of 
settled bed - kg/m . 

Reading stops when IT(J) < 0 

SET H INITIAL DEPTHS OF FLOW 

Depths of flow at each node are read in depending on the value of 
IDEP. Read from file unit INH 

IDEP = 0 All depths set to 1.0 m by default. 

IDEP = 1 All nodal depths set to constant. 

CARD H.l (Flo. 5) Constant value of depth. 

1-10 : CDEP Depth of flow - m 

IDEP = 2 Read node point depths from file INH. 

CARD H.2 (4(110, FIO. 5)) 

1-10 : IT(J) Node number 

11-20 : TEMP(J) Depth of flow - m 

Stops reading if IT(J) < 0 



319 

IDEP = 3 Compute depths from user supplied procedure in 
Subroutine DEPTH. 

SET I INITIAL VELOCITY FIELD 

The horizontal velocity components at each node must be specified. The 
value of IVEL determines type of input. This input only for unsteady 
problems. All reads are from file number INF. 

IVEL = 1 Velocities are set to constant values. 

Read from file unit INF 

CARD I.l (2F10.5) 

1-10 : CONXY Velocity ' component in the x-di recti on - 

m/s 

11-20 : CONYV Velocity component in the y-direction - 

m/s 

IVEL = 2 Each nodal velocity component read in. 

CARD 1.2 (4(2F10.5)) Must be read in order for all NP nodes. 

1-10 : XVEL(J,1) Velocity component at node J in the x- 

di recti on - m/s 

11-20 : XVEL{J,2) Velocity component at node J in the y- 

di recti on - m/s 

IVEL = 3 User supplied procedure in Subroutine VEL is 

used to calculate nodal velocities. 

SET J INITIAL DISPERSION COEFFICIENTS 

The form of input is set by the value of IDIF, 

IDIF = 1 Dispersion coefficients are set to constant values. 

CARD J.l (2F10.5) 

1-10 : CDIFL Longitidinal dispersion coefficient - 

m^/s 

11-20 : CDIFT Transverse dispersion coefficient - 

m^/s 

IDIE = 2 Dispersion coefficients are read in node by node. 

CARD J. 2 et.seq.(3(I5,2F10.5) 

1-5 : IT(J) Node number 

6-15 : TEMP(1,J) Longitudinal dispersion coefficient - 

m2/s 

16-25 : TEMP(2,J) Transverse dispersion coefficient - 

".2/s 



m' 



320 



Reading stops for IT(J) < 0 

IDIF = 3 Dispersion coefficients D^^^, D^^ , 0^^^ and D 

computed analytically using dispersion algorithm. 

For NOPT = 3 : 

SET K INITIAL SETTLING VELOCITIES 

The initial settling velocities at each node point must be read in. 
The form of input is determined by the value of I SET. 

Read from file unit ING 

ISET = 1 All settling velocities are set to constant. 
Card K.l {F10.5) 

1-10 : CVSX Settling velocity - m/s 

ISET = 2 

CARD K.2 et.seq.(4(I10,F10.5)) 

1-10 : IT(J) Node number 

11-20 : TEMP(J) Settling velocity - m/s 

Stops reading if IT(J) < 0 



ISET = 3 Settling velocity model, for which parameters were 
read in SET D, is used to compute each nodal 
settling velocity. 



SET L BOUNDARY CONDITIONS 

For any problem, concentration boundary conditions must be specified 
at least at one node. At all external boundaries that have no 
concentration specified, the normal diffusive flux is defaulted to zero. 
Type of input is determined by value of IN8C. File number for input is INB. 

INBC = 1 Read node number and specified boundary condition from 

cards. 

CARD L.l (3(110, FIO. 5)) 

1-10 : IT(J) Node number 

11-20 : TEMP(J) Specified concentration - kg/m^ 

For J=1,...,NP 



INBC = 2 Concentration computed according to user supplied 
procedure in Subroutine CONCBC 



321 



I NBC = 3 Read node number and specified boundary condition from 
tape file 1MB. 

CARD L.2 

MFIX(J) Equal to 1 for boundary node; 

0 for all other nodes 
SPEC(J,1) Specified concentration at node J - kg/m"^ 

For J=1,...,NP 

SET M NEW SALINITIES 

Type of input is determined by value of ISALC{J), for J=2, . . . ,NTTS. 
ISALC(J)=1 

CARD M.l (F10.5) 

1-20 : SVI Constant salinity value read in - ppt 

ISALC(J)=2 File number for input - INSS 

CARD M.2 (3(110, FIO. 5)) Salinities for specified nodes 

1-10 : IT(J) Node number 

11-20 : TEMP (J) New salinity at node J - ppt 

Reading stops where IT(J) < 0. 

ISALC(J)=3 File number for input - INSS 

CARD M.3 (7F10.5) New salinities for all nodes. 

1-10 : SAL(J) Salinity at Jth node 



for J=1,...,NP 



DYNAMIC INPUT 



The same subroutines that read initial values are used to read changes 
in these values during a dynamic run. The input code arrays in SET C 
tell the program if any new values should be read in at each time step. 
Note that the starting time is time step 1. The order of reading each set 
of data is given below. If the code is zero, no input is required. If it 
is non-zero, the value of the code will determine the type of input. 



322 



DESCRIPTION 



CODE ARRAY 



INPUT CARD SET 



Salinities ISALC(J) 

Concentration B.C. IFF(J,1) 

Depths of flow IDEPC(J) 

Velocities IVCOD(J) 

Dispersion coefficients IDIF(J) 

Settling velocities ISVS(J) 



SET M 
SET L 
SET 
SET 
SET 
SET 



I 

F 
E 
J 



only for sediment problems 



APPENDIX D 
DATA COLLECTION AND ANALYSIS PROGRAMS 



D.l. Field Data Collection Program 

A field data collection program for a sedimentation study should 
consist of four principal components: 1) hydrographic survey, 2) 
sediment sampling, 3) measurement of suspended sediment concentration, 
water temperature and salinity and 4) determination of sediment settling 
velocity. The collection program required for modeling the hydrodynamic 
regime in an estuary is rather well known, and will not be addressed. 
Hydrographic Survey 

At least two sonar fathometers and a gamma-ray transmission 
densitometer should be used to measure the depths in the water body to 
be modelled. First it is recommended that the entire water body be 
surveyed simultaneously using, for example, a 30-kHz and a 200-kHz 
fathometer. The Kelvin Hughes Division of Smiths Industries Ltd. (U.K.) 
manufactures a 30-kHz fathometer (model MS48) that has an approximate 
maximum range and resolution of 1000 m and 10 mm on the 0-20 range, 
respectively. Raytheon (USA) manufacturers a 200-kHz fathometer with an 
approximate maximum range and resolution of 120 m and 0.5% of depth, 
respectively. The 200-kHz fathometer has a maximum range that is 
approximately one order of magnitude less than that of the 30-kHz 
instrument because of the greater attenuation of the high frequencies in 
water. Raytheon does manufacture a dual 22.5-kHz and 200-kHz frequency 

fathometer. No specifications are available for this model. 

323 



324 



Parker and Kirby (1977) reported that the sediment-water interfaces 
of stationary suspensions in the Severn Estuary. England were detected 
by a 200-kHz fathometer but not by a 30-kHz fathometer. In areas where 
stationary suspensions are determined to exist (by comparison of the 30- 
kHz and 200-kHz records), it is recommended that a gamma-ray 
transmission densitometer or a turbidity meter be used to supplement the 
depth record obtained with the fathometers. A fathometer alone may not 
be capable of detecting the surface of a stationary suspension for the 
following reasons: 1) The acoustic detection of a dense suspension 
depends on the gradient of the bulk density at the surface of the 
suspension and not on the magnitude of the density. In mobile 
suspensions and in newly formed stationary suspensions this density 
gradient is very small, and in most cases the surfaces of these 
suspensions will not be detectable. 2) As the stationary suspension 
undergoes consolidation, different parts (i.e. levels) of the suspension 
may become detectable to fathometers with different frequencies at 
different times, which makes the interpretation of such fathometer 
records a difficult and uncertain task (Parker and Kirby, 1977). 

A gamma-ray transmission densitometer obtains in situ measurements 
of the sediment bulk density profiles, and thus can be used, in 
addition, to determine the thickness of stationary suspensions and the 
location (i.e. vertical elevation with respect to geodetic datum) of the 
top of the settled bed, at which the bulk density is usually assumed to 
be 1300 kg/m . A static cone penetrometer directly measures penetration 
resistance and indirectly measures the shear strength of the sediment. 
The densitometer has to be calibrated at the beginning and end of survey 
operations to determine the relationship between the radiation count 



325 

rate and sediment density. Calibration is generally performed using 
liquids with different densities. The densitometer is penetrated and 
retracted in the sediment at a rate of approximately 2 to 3 mm/sec, 
during which the radiation count rate and probe penetration depth are 
continuously recorded. This system has been repeatedly used from both 
ships and submersibles and has the capability of measuring in situ bulk 
sediment densities up to 1800 kg/m3. operating in depths up to 3.6 km. 
and penetrating one to two meters in cohesive sediment suspensions 
(Hirst etal_^, 1975). Harwell, the United Kingdom Atomic Energy 
Research Establishment, manufacturers a transmission densitometer that 
has a vertical resolution of ± 1 cm with an accuracy of ± 2% in the bulk 
density range 1000-2000 kg/m^ (Parker etal_., 1975). 

There are three methods used to fix the boat position during a 
hydrographic survey: 1) optical methods. 2) electronic methods and 3) 
combined systems. Optical methods include double horizontal sextant 
angle resection, single horizontal sextant angle and transit line, and 
theodolite intersections from shore. Electronic methods include two 
megahertz systems, microwave systems and range and bearing systems, 
while combined systems use a theodolite to determine the bearing and a 
microwave system to determine the distance. A description of these 
three methods is given by Ingham (1975) and Dyer (1979). 
Sediment Sampling Using Corers 

Before the data collection period begins, at least two 10-12 cm 
diameter cores should be collected at each sampling station. There are 
four types of corers. These are the gravity-corer, piston-corer, 
vibracorer and box-corer. A gravity corer is lowered close to the ocean 
floor and a tripping mechanism is released so that the last part of the 



326 

descent is in free fall. The core barrel is supposed to penetrate the 
sediment, cutting out a cylinder of mud. The barrel is equipped with a 
plastic liner which can be slipped out. An orange peel core catcher is 
located between the nose piece and the liner to prevent the core from 
washing out when the corer is retrieved. In general, gravity corers 
have a barrel length of 1 to 2 m. 

A piston corer is released and free falls a known distance as soon 
as a tripping weight hanging a known distance below the corer senses the 
bottom. During penetration of the corer in the sediment the piston 
moves up the core liner. This action permits the hydrostatic pressure 
head of the water column to aid the corer in penetrating the sediment 
and removing the water from the barrel to reduce the resistance to the 
core as it enters the liner. Cores in excess of 20 m have been obtained 
from soft sediments. 

A vibrator mechanism is used to drive a vibracorer into sediment. 
In general, this type of corer is heavy and requires a large vessel for 
operation. However, a lighweight pneumatic corer has recently been 
designed and field tested at the U.S. Army Corps of Engineers Coastal 
Engineering Research Center (CERC) (Fuller and Meisburger, 1982). This 
system can be used from relatively small vessels for obtaining 5.0 cm 
diameter cores of unconsolidated sediments from 0.6 to 2.4 m in length. 

A box-corer is used to obtain large cores (up to 50 cm in height, 
30 cm in length and 20 cm in width) of sediment when it is essential 
that the internal fabric of the sediment not be disturbed. Box-corers 
consist of an open steel box that is driven vertically into the sediment 
until the top of the box rests on the surface of the sediment. The open 
end of the box is covered by a blade which cuts through the sediment. 



327 

This type of corer has been used to obtain excellent cores of estuarine 
sediment (Dyer 1979). 

Measurement of Suspen ded Concentration. Salinity and Temperature 

The first item that must be considered is the time period over 
which data will be collected for eventual use in the model. This will 
be contingent upon the desired results from the modeling effort. In 
tidal water bodies, data should preferably be collected over a minimum 
of 15 hours (assuming the tide is semi-diurnal) over three different 
tidal cycles: spring, mean and neap. It would be more desirable to 
have the data collection period span at least one week, starting, for 
example, on a spring tide and finishing at the subsequent neap tide. 
The next consideration is the number of data sampling stations and where 
they should be located in order to adequately monitor the spatial 
variations of the concentration of suspended sediment. Stations must be 
located at all exterior water boundaries (cross-sections) of the 
estuarial system to be modeled. The width of the boundary cross-section 
and the lateral variability of the depth should be considered when 
deciding upon the minimum number of stations to be located laterally 
across such a boundary. For example, stations would definitely be 
located at predominant features such as navigation channels. Additional 
stations must be located at all interior confluences and bifurcations, 
and at as many other interior locations as possible. It is recognized 
that the length of the data collection period and the number of stations 
are often less than desired due to economic and logistical 
considerations. 

At each station the location of the top of the sediment bed with 
respect to a geodetic or tidal datum must be determined using the 



328 

previously described surveying methods. It is recommended the water 
temperature, electrical conductivity (or salinity) and concentration of 
suspended sediment be measured at least once every one-half hour for the 
duration of the collection period at each of the sampling stations. 
These measurements should preferably be made at a minimum of three 
depths over the vertical: one-half meter below the water surface, mid- 
depth and one-half meter above the bottom (i.e. top of sediment bed). 
For locations where the water depth is greater than about 3 to 4 m, 
measurements should be made at additional depths over the vertical. 
Both the measurement and analysis of water temperature and electrical 
conductivity data are discussed by Dyer (1979). A description of 
various filtration procedures for determining the suspension 
concentration gravi metrically is given by Dyer (1979). 

There are three general methods used to measure the suspension 
concentration: 1) water sampling, 2) optical methods and 3) gamma-ray 
densitometer measurements. 

Water bottles and shipboard pumps are the two most common water 
sampling devices. The NIO bottle has capacities from 1.25 to 7.1 
liters. Other water bottles, such as the Van Dorn bottle, have 
capacities of up to 10 liters or more. The NIO bottle consists of a PVC 
tube open at both ends with hemispherical bings on spring loaded arms 
which close each end. Most water bottles are closed by dropping a brass 
messenger down the support wire (Dyer, 1979). 

Shipboard pumps are used to pump water samples up to the vessel 
through an intake tube mounted on an instrument package. It is 
recommended that in situ separation of the water and sediment be 
performed on the vessel using the filter method (van Rijn, 1979). 



329 

Instruments for optical determination of the concentration of 
suspended sediment include the transmissometer, nephelometer and the 
Secchi disc. Nephelometers are not very practical for use in 
estuaries since they are sensitive to very low concentrations only. 
Secchi discs can be used to estimate surface values only. 
Transmissometers, or electro-optical turbidity meters, have been used 
successfully to measure vertical turbidity profiles in, among others, 
the Severn, Maas, James and Rappahannock estuaries and in Upper 
Chesapeake Bay (Kirby and Parker, 1977; Nichols ^al_., 1979). These 
meters can be used to detect both mobile and stationary suspensions as 
their operating range, in general, is 0.25-25 g/1 . They have a rapid 
response time (100 Hz) which allows profiles in 30 m depths to be taken 
in 15-20 seconds. Partech (U.K.) is one manufacturer of optical 
turbidity meters. 

Both Kirby and Parker (1977) and Nichols etaj_. (1979) used 
instrument arrays on which were mounted a gamma-ray transmission 
densitometer, at least one electro-optical turbidity meter, an 
electromagnetic current meter, a pressure transducer and a water 
temperature and conductivity probe. In general, the optical turbidity 
meters would be used to record the concentration profile for suspensions 
up to 25 g/1 and the transmission densitometer used for denser 
suspensions. 

Determination of Sediment Settling Velocity 

An appropriate method to measure settling velocities is by using an 
instrument similar to the sampling tube developed by Owen (1971), in 
which undisturbed samples of suspended sediments are collected in situ 
in their natural state. The settling velocities of the aggregates are 



330 

determined immediately thereafter through use of a bottom withdrawal 
sedimentation test. Allersma (1980) gives a detailed description of an 
I'n situ suspended sediment sampler. 

D.2 Laboratory Sediment Testing Program 

It is recommended that the following physiochemical sediment and 
fluid properties be determined using the collected sediment cores. 
Properties of Undisturbed Sediment Cores 

The gamma-ray densitometer may be used to determine the bulk 
sediment density profile in the undisturbed cores still in the liner 
tubes as soon after the cores are obtained as possible. A description 
of this procedure is given by Whitmarsh (1971) and Kirby and Parker 
(1974). If this instrument is not available, the freeze-drying 
procedure used by Parchure (1980) and Dixit (1982) or the pumping method 
used by Thorn and Parsons (1977) may be used to determine the bulk 
density profile. The pumping method consists of removing by suction a 
thin layer, e.g. 3 cm, from the top of the core. This procedure is 
repeated, layer by layer, and each layer is analyzed to determine the 
mean bulk density. 
Properties of Original Settled Bed 

The bulk density and bed shear strength profiles and the erosion 
rate constant for each layer need to be determined for the cores. The 
number of layers and the thickness of each are determined from the 
nature of the bed shear strength profile. The erosion rate constant for 
each layer and the shear strength profile can be determined, for 
example, in the rotating cylinder erodibility testing apparatus 
described by Sargunam et al_. (1973). In order to use this apparatus. 



331 

the core sample must be trimmed. The portion of each core that is 
sufficiently consolidated such that it can be trimmed and tested in the 
erosion apparatus is defined to be the settled bed. The thickness of 
this portion defines the location of the top of the settled bed. Soft, 
unconsolidated portions of each core are assumed to be new deposits. 
Properties of New Deposits 

For cores with soft, unconsolidated or partially consolidated 
sediment on top of the settled portion, the following method may be used 
to estimate the erosional, depositional and consolidation 
characteristics of such new deposits. The new deposit samples from the 
cores at all the stations should be mixed and subjected to laboratory 
erosion, deposition and consolidation tests (described by Parchure 
(1980), Mehta and Partheniades (1973), in Sections 3.4 and 3.6 and 
below) to determine: the settling velocity as a function of suspension 
concentration and salinity; the minimum and maximum depositional shear 
stresses -c^^:^^ and -z^^^^; variation of tgg and with the bed shear 
stress, -v^; the number of characteristic stationary suspension layers, 
and the thickness, dry sediment density and shear strength of each 
layer; the number of characteristic partially consolidated new deposit 
layers, and the thickness, dry sediment density, shear strength and the 
resuspension parameters and a of each layer; the variation of 9^ and 
Tdc„ with C^; variation of p(z) with T^^; variation of the bed shear 
strength, t^, with p. The variation of the bed density and shear 
strength profiles with salinity can be determined by performing the 
erosion tests at several salinities between 0 and 35 ppt. A brief 
description of laboratory tests which can be conducted in order to 
determine the above mentioned consolidation parameters is given next. 



332 

Laboratory tests to determine the consolidation characteristics of 
a cohesive sediment bed involve the measurement of the bed density 
profile. Various methods have been used for this purpose. Been and 
Sills (1931) measured the density profile of a clayey soil using a non- 
destructive X-ray technique. Methods which involve the destruction of 
the soil include, among others, the freeze-drying procedure used by 
Parchure (1980) and Dixit (1982), the pumping, or layer-by-layer 
sampling method used by Thorn and Parsons (1977), and the use of 
specially designed apparatuses (Parchure, 1980). The latter consists of 
a 183 cm high, 30 cm diameter PVC cylinder, a bottom plate, and ten 1.27 
cm diameter plastic tubes ranging from 1.27 to 12.7 cm in height glued 
to the bottom plate, concentric to the PVC cylinder. These cylinders 
are filled with a sediment suspension of known concentration, and the 
sediment is allowed to settle under quiescent conditions for a specified 
consolidation time. Following the procedure described by Parchure 
(1980), the bulk density profile can be determined. 

The following parameters should be varied systematically in the 
laboratory tests in order to determine their effect on the rate of 
consolidation: 

(a) Consolidation time - it is recommended that the time allowed for 
the bed to consolidate before the density profile is measured be 
varied logarithmically from 0 to 720 hours (1 month). 

(b) Initial conditions - the initial suspension concentration, which 
determines the thickness and density of the initial bed. 

(c) Salinity - the salinity of the water should be varied from 0 (tap 
water) to 35 ppt. 



333 

(d) Overburden - it is important that the effect of discretized to 

continuous additions of varying amounts of sediment (overburden) on 
top of the initial bed be determined in order to evaluate the 
effect of such overburden pressures on the consolidation rate of 
the lower bed layers. 

The relationship between p and needs to be determined as well 
for the collected sediment samples. Both the bed shear strength and 
density profiles may be determined using the methodology described by 
Mehta et al. (1982a). These profiles can then be used to establish an 
empirical relationship between and p. 
Fluid Composition 

The pH, total salt concentration, and concentrations of ions such 
as Na"*", Ca^"^, Mg^"^, K"^, Fe^"^ and CI" should be determined for both the 
pore fluid in the consolidated bed portion of one core and a sample of 
the suspending fluid. 

Composition and Cation Exchange Capacity of the Sediment 

The sediment contained in the consolidated bed portion of one core 
from each collection station should be thoroughly mixed so that a 
spatially homogeneous sample is obtained. A standard hydrometer 
analysis must be conducted on each so-prepared sample to determine the 
sediment particle size distribution and thereby the percentage by weight 
of clay, silt and fine to coarse sand in each sample. In preparing the 
samples for this analysis, the sediment must not be initially air-dried 
(to obtain the dry weight of the material used in the test), as it has 
been found that dried sediment will not completely redisperse when the 
dispersing agent is added (Krone, 1962). For this reason, the total dry 
weight of the sample must be obtained after the test by evaporating off 



334 

all the water in an oven set at approximately 50° C. The percentage of 
weight of organic matter should be determined through use of a method 
such as the Walkley-Black test (Allison, 1965). In addition, it is 
recommended that X-ray diffraction analysis of the bulk sample, and < 2 
m unglycolated and glycolated portions be conducted in order to 
determine the predominant clay and non-clay mineral constituents. The 
cation exchange capacity must be determined for each sample. 



335 



REFERENCES 



Aitchison, J., and Brown, J. A. C, The Lognormal Distribution . 

Umversity of Cambridge, Dept. of Applied Economics, Monograph No 5 
Cambridge, United Kingdom, 1957. 

Allersma, E., "Mud in Estuaries and Along Coasts," Interna tional 
^y^Posium o n River Sedimentation. Beijing, P. R. of China, March, 

Alizadeh, A., "Amount and Type of Clay and Pore Fluid Influences on the 
Critical Shear Stress and Swelling of Cohesive Soils," Ph.D 
Dissertation. University of California, Davis, California! 1974. 

Allison, L. E., Organic Carbon, Part II. American Society of Aqronomv, 
Madison, Wisconsin, 1965, 

Alonso, C.V., "Stochastic Models of Suspended-Sediment Dispersion " 
Journal of the Hydraulics Division. ASCE, Vol. 107, No. HY6, June. 
1981, pp. 733-757. 

Ariathurai, R., "A Finite Element Model for Sediment Transport in 
Estuaries," Ph.D. Dissertation. University of California, Davis 
California, 1974. 

Ariathurai, R., and Krone, R. B., "Finite Element Model for Cohesive 
Sediment Transport," Journal of the Hydrau lics Division, ASCE 
Vol. 102, No. HY3, March, 1976, pp. 323-338. 

Ariathurai, R., MacArthur, R. C, and Krone, R. G., "Mathematical Model 
of Estuanal Sediment Transport," Technical Report D-77-12 . U.S. Army 
Engineers Waterways Experiment Station, Vicksburg, Mississipoi 
October, 1977. 

Ariathurai, R., and Arulanandan, K., "Erosion Rates of Cohesive Soils " 
Journal of the Hydraulics Division. ASCE, Vol. 104, No. HY2 
February, 1978, pp. 279-283. 

Ariathurai, R., and Mehta, A.J., "Fine Sediments in Waterway and Harbor 
Shoaling Problems," International Conference on Coastal and Port 
Engineering in De veloping Countries , Colombo. Sri Lanka. Marrh, IQP? 

Aris, R., "On the Dispersion of a Solute in a Fluid Flowing Through a 
Tube, Proc. Royal Society of Lon don. Series A. Vol. 235 1956 
pp. 67-77; ' ' 

Arulanandan, K., "Fundamental Aspects of Erosion of Cohesive Soils " 
Journal of the Hydraulics Division. ASCE, Vol. 101, No. HY5, March, 
1975, pp. 635-639. 



336 



Arulanandan, K., Loganathan, P., and Krone, R. B., "Pore and Eroding 
Fluid Influences on Surface Erosion of Soil," Journal of Geotechnical 

Engineering Division , ASCE, Vol. 101, No. GTl, January, 1975, 

pp. 51-66. 

Arulanandan, K., Sargunam, A., Loganathan, P., and Krone, R. B., 

"Application of Chemical and Electrical Parameters to Prediction of 
Erodibility," Special Report 135 . Highway Research Board, Washington. 
D.C., 1973, pp. 42-51. 

Bain, A. J., "Erosion of Cohesive Muds," M.S. Thesis , University of 
Manchester, United Kingdom, 1981. 

Barnes, R,. S. K., and Green, J., Editors, The Estuarine Environment , 
Applied Science Publishers, London, 197TT ~ 

Bauer, L., "Water Quality and Heavy Metals in Sediments of Thirty-Six 
Florida Marinas," M.S. Project Report, Dept. of Environmental 
Engineering Sciences, University of Florida, Gainesville, 1981. 

Been, K., and Sills, G. C, "Self-weight Consolidation of Soft Soils: 
An Experimental and Theoretical Study," Geotechnique, Vol. 31, No. 4, 
1981, pp. 519-535. 

Bellessort, B., "Movement of Suspended Sediments in Estuaries- 
Flocculation and Rate of Removal of Muddy Sediments," in Tracer 
Techniques in Sediment Transport . International Atomic Energy Agency, 
Technical Report Series No. 145, 1973, pp. 31-40. 

Beltaos, S., "Longitudinal Dispersion in Estuaries," Journal of the 
Hydraulics Division, ASCE, Vol. 106, No. HYl, January, 1980a, 
pp. 151-172. 

Beltaos, S., "Transverse Mixing Tests in Natural Streams," Journal of 
the Hydraulics Division. ASCE, Vol. 106, No. HYIO, October, 1980b. 
pp. 1607-1625. 

Bolz, R. E. and Tuve, G. L., Editors, Handbook of Tables for Applied 
Engineering Science. Chemical Rubber Co., Cleveland, Ohio, 1976. 

Cargill, K.W., "Consolidation of Soft Layers of Finite Strain Analysis," 
MPGL-82-3. Geotechnic Laboratory, U.S. Army Engineer Waterways 
Experiment Station, Vicksburg, Mississippi, 1982. 

Chatwin, P.C., "Presentation of Longitudinal Dispersion Data," Journal 
of the Hydraulics Division. ASCE, Vol. 106, No. HYl, January, 1980, 
pp. 71-83. 

Chen, C, "Sediment Dispersion in Flow with Moving Boundaries," Journal 

of the Hydraulics Division. ASCE, Vol. 97, No. HY8, August. 1971^ 

pp. 1181-1201. ~ 

Christensen, B. A., Discussion of "Erosion and Deposition of Cohesive 
Soils," by Partheniades, E., Journal of the Hydraulics Divisio n. 
ASCE, Vol. 91, No. HY5, September, 1965, pp. 301. 



337 

Chrlstensen. B. A., "Sediment Transport," Unpublished Lecture Notes 
University of Florida, Gainesville, Florida, May, 1977. ' 

Christensen, R. W., and Das, B. M., "Hydraulic Erosion of Remolded 
Cohesive Soils," Special Report 135. Highway Research Board 
Washington, D. C., 1973, pp. 8-19. 

Cole, P., and Miles, G. v., "Two-Dimensional Model of Mud Transport " 
Journal of Hydraulic Engineering Divis ion. ASCE, Vol. 109 No l' 
January, 1983, pp. 1-12. * ' 

Croce, P., "Evaluation of Consolidation Theories by Centrifugal Model 
Tests, M.S. Thesis, University of Colorado, Boulder, Colorado, 1982. 

Dagan, G., "Dispersi vity Tensor for Turbulent Uniform Channel Flow " 
Journal of the Hydraulics Division. ASCE, Vol. 95, No. HY5 
September, 1969, pp. 1699-1712. 

Dixit, J G., "Resuspension Potential of Deposited Kaolinite Beds," 
M.S. Thesis, University of Florida, Gainesville, Florida, 1982. 

Dixit, J. G., Mehta, A. J., and Partheniades, E., "Redeposi tional 
Properties of Cohesive Sediments Deposited in a Long Flume," 
UFL/COEL-82/002. Coastal and Oceanographic Engineering Department, 
University of Florida, Gainesville, Florida, August, 1982. 

°^^I"o7^ ^" Estuaries; A Physical Introduction. John Wiley, London, 
1973. 

Dyer, K. R., Editor, Estuarine Hydrography and Sedimentati on. Cambridge 
University Press, Cambridge, United Kingdom, 1979. 

Edzwald, J. K., Upchurch, J. B., and O'Melia, C. R., "Coagulation in 
Estuaries, Environmental Science & Technology. Vol. 8 No 1 
January, 1974, pp. 58-63. ' ^ * ' 

Einsele. G., Overbeck, R., Schwarz, H. U., and Unsold, G., "Mass 
Physical Properties, Sliding and Erodibility of Experimentally 
Deposited and Differently Consolidated Clayey Muds," Sedimentology, 
Vol. 21, 1974, pp. 339-372. " ^ 

Einstein, H. A., and Krone, R. B., "Experiments to Determine Modes of 
Cohesive Sediment Transport in Salt Water," Journa l of Geophysical 
Reseai^ch . Vol. 67, No. 4, April, 1962, pp. 1451-1464. 

Elder, J. W., "The Dispersion of Marked Fluid in Turbulent Shear Flow " 
Journal of Fluid Mechanics . Vol. 5, 1959, pp. 544-560. 

Fischer, H. B., "Longitudinal Dispersion in Laboratory and Natural 
Sl^i^eams,' Ph.D. Dissertation. California Institute of Technology. 
Pasadena, California, 1966. 

Fischer, H. B., Discussion of "Dispersi vity Tensor for Turbulent Uniform 
Channel Flow," by G. Dagan, Journal of the Hydraul ics Division, ASCE, 
Vol. 96, No. HY4, April, 1970, pp. 1096-1100. 



338 



Fischer, H, B., "Mass Transport Mechanisms in Partially Stratified 
Estuaries," Journal of Fluid Mechanic s, Vol. 53, Part 4, 1972 
pp. 671-687. ' — ~ 

Fischer, H. B., "On the Tensor Form of the Bulk Dispersion Coefficient 
in a Bounded Skewed Shear Flow," Journal of Geophysi cal Research 
Vol. 83, Mo. C5, May, 1978, pp. 2373-2375. ' 

Fischer, H. B., Imberger, J., List, E. J., Koh, R. C. Y., and Brooks, N. 
H., Mixing in Inland and Coastal Haters . Academic Press, New York. 

1 ft "7 n ' * 



Fuller, J„ A., and Meisburger, E. P., "A Lighweight Pneumatic Coring 
Device: Design and Field Test," Miscellaneous Report No. 82-8 . U.S. 
Army Corps of Engineers Coastal Engineering Research Center, Fort 
Belvoir, Virginia, September, 1982. 

Gibson, R. E., England, G. L., and Hussey, M. J. L., "The Theory of One- 
Dimensional Consolidation of Saturated Clays, I. Finite Nonlinear 
Consolidation of Thin Homogeneous Layers," Geotechnique. Vol 17 
1967, pp. 261-273. — 

Gibson, R. E., Schiffman, R. L., and Cargill, K. W., "The Theory of One- 
Dimensional Consolidation of Saturated Clays, II. Finite Nonlinear 
Consolidation of Thick Homogeneous Layers," Canadian Geotechnical 
Journal. Vol. 18, 1981, pp. 280-293. " 

Glenne B., and Selleck, R. E., "Longitudinal Estuarine Diffusion in San 
Francisco Bay, California," Water Research. Vol. 3, 1969, pp. 1-20. 

Grim, R. E., Applied Clay Mineralogy . McGraw-Hill, New York, 1962. 

Grim, R. E., Clay Mineralogy. McGraw-Hill, New York, 1968. 

Grimshaw, R. W., The Chemistry and Physics of Clays . Wiley-Interscience. 
New York, 1971"^ ~ 

Gularte, R.C., "Erosion of Cohesive Sediment as a Rate Process," 
Ph.D. Dissertation. University of Rhode Island, 1978. 

Gust, G., "Observations on Turbulent-drag Reduction in a Dilute 
Suspension of Clay in Sea-water," Journal of Fl uid Mechanics, 
Vol. 75, Part 1, 1976, pp. 29-47. 

Hanzawa, H., and Kishida, T., "Fundamental Considerations of Undrained 
Strength Characteristics of Alluvial Marine Clays," Soils and 
Foundatons . Japanese Society of Soil Mechanics and Foundation 
Engineering, Vol. 21, No. 1, March, 1981, pp. 39-50. 

Hayter, E.J., and Mehta, A.J., "Modeling of Estuarial Fine Sediment 
Transport for Tracking Pollutant Movement," UFL/COEL-82/009. Coastal 
and Oceanographic Engineering Department, University of Florida, 
Gainesville, Florida, December, 1982. 



339 



Hirst, T. J., Perlow, M. Richards, A. F., Burton, B. S., and van Sciver, 
W. J., "Improved In Situ Gamma-Ray Transmission Densitometer for 
Marine Sediments.' ^Ocean Engineering. Vol. 3, No. 1, 1975, pp. 17-27. 

Holley, E. R., "Unified View of Diffusion and Dispersion," Journal of 
the Hydraulics Division. ASCE, Vol. 95, No. HY2, March, 1969, 
pp. 621-631. 

Holley, E. R., Harleman, D. R. F., and Fischer, H. B., "Dispersion in 
Homogeneous Estuary Flow," Journal of the Hydraulics Division. ASCE, 
Vol. 96, No. HY8, August, 1970, pp. 16^1-l7M. 

Hood, P., "Frontal Solution Program for Unsymmetric Matrices," 

International Journal for Numerical Met hods in Engineering, Vol. 10, 
1976, pp. 379-39^. ^ ^ 

Huang, S., Han, H., and Zhang, X., "Analysis of Siltation at Mouth Bar 
of the Yangtze River Estuary," Prbc. International Conference on 
River Sediment Transportation. Peking, China, March, 1980. 

Hughes, P„ A., "A Determination of the Relation Between Wind and Sea- 
Surface Drift," Quarterly Journal of the Royal Meteorological 
Society . London, Vol. 82, October, 1956, pp. 494-502. ' 

Hunt, J. R., "Prediction of Oceanic Particle Size Distribution from 
Coagulation and Sedimentation Mechanisms," In Advances in Chemistry 

Series No. 189 - Particles in Water. M. D. Kavanaugh and J. 0. 

Keckie, Editors, American Chemical Society, 1980, pp. 243-257. 

Hunt, S. D., "A Comparative Review of Laboratory Data on Erosion of 
Cohesive Sediment Beds," M.E. Project Report. University of Florida, 
Gainesville, Florida, 1981. 

Ingham, A. E., Editor, Sea Surveying. John Wiley & Sons, Chichester, 
United Kingdom, 1975. 

Ippen, A. T., Editor, Estuary and Coastline Hydrodynamics. McGraw-Hill. 
New York, 1966. — 

Jobson, H. E., and Sayre, W. W., "Vertical Transfer in Open Channel 
f^low," Journal of the Hydraulics Division . ASCE, Vol. 96, No. HY3, 
March, 1970, pp. 703-724. 

Kandiah, A., "Fundamental Aspects of Surface Erosion of Cohesive Soils," 
Ph.D. Dissertation. University of California, Davis, California. 
1974. 

Karcz, I., and Shanmugam, G., "Decrease in Scour Rate of Fresh Deposited 
Muds," Journal of the Hydraulics Division. ASCE, Vol. 100, No. HYll, 
November, 1974, pp. 1735-1738. 

Kelley, W. E., and Gularte, R. C., "Erosion Resistance of Cohesive 

Soils," Journal of the Hydraulics Division. ASCE, Vol. 107, No. HYIO. 
October , 1981. 



340 



King, I. P., Norton, W. R., and Orlob, G. T., A Finite Element Solution 
for Two-Dimensional Density Stratified Flow, U.S. Department of the 
Interior, Office of Water Resources Research, April, 1973. 

Kirby, R., and Parker, W. R., "Fluid Mud in the Severn Estuary and the 
Bristol Channel and Its Relevance to Pollution Studies," Proc. 
International Chemical Engineers Exeter Symposium , Exeter, United 
Kingdom, Paper A-4, 1973, pp. 1-14. 

Kirby, R., and Parker, W. R., "Seabed Density Measurements Related to 
Echo Sounder Records," Dock and Harbour Authority , Vol. 54, 1974, 
pp. 423-424. 

Kirby, R., and Parker, W. R., "The Physical Characteristics and 
Environmental Significance of Fine Sediment Suspensions in 
Estuaries," in Estuaries. Geophysics and the Environment , National 
Academy of Sciences, Washington, 1977, pp. 110-120. 

Kirby, R., and Parker, W. R., "Distribution and Behavior of Fine 
Sediment in the Severn Estuary and Inner Bristol Channel, U.K.," 
Canadian Journal of Fisheries and Aquatic Sciences , Vol. 40, 
Supplement Number 1, 1983, pp. 83-95. 

Koutitas, C, and O'Connor, B. A., "Numerical Modelling of Suspended 
Sediments," Advances in Water Resources, Vol. 3, June, 1980, 
pp. 51-56. 

Kranck, K., "Sedimentation Processes in the Sea," in The Handbook of 
Environmental Chemistry, Vol. 2, Part A, 0. Hutzinger, Editor, 
Springer-Verlag, Berlin, 1980, pp. 61-75. 

Krone, R. B., "Flume Studies of the Transport of Sediment in Estuarial 
Shoaling Processes," Final Report , Hydraulic Engineering Laboratory 
and Sanitary Engineering Research Laboratory, University of 
California, Berkeley, California, June, 1962. 

Krone, R. B., "A Study of Rheological Properties of Estuarial 

Sediments," Technical Bulletin No. 7, Committee of Tidal Hydraulics, 
U.S. Army Corps of Engineers, Vicksburg, Mississippi, September, 
1963. 

Krone, R. B., "A Field Study of Flocculation as a Factor in Estuarial 
Shoaling Processes," Technical Report No. 19, Committee on Tidal 
Hydraulics, U.S. Army Corps of Engineers, Vicksburg, Mississippi, 
June, 1972. 

Krone, R. B., "Aggregation of Suspended Particles in Estuaries," in 
Estuarine Transport Processes , P. Kjerfve, Editor, the Belle W. 
Baruch Library in Marine Science, No. 7, University of South Carolina 
Press, Columbia, South Carolina, 1978, pp. 177-190. 

Kruyt, H. R., Editor, Colloid Science , Vol. I, Elsevier, Amsterdam, New 
York, 1952. 



341 



Kuo, A., Michols, M., and Lewis, J., "Modeling Sediment Movement in the 
Turbidity Maximum of an Estuary," Bulletin 111, Virginia Institute of 
Marine Science, Gloucester Point, Virginia, 1978. 

Lambe, T. W., "The Structure of Inorganic Soil," Transacti ons of the 
ASCE , Vol. 79, Mo. 315, 1953. 

Lambermont, J., and Lebon, T., "Erosion of Cohesive Soils," Journal of 
Hydraulic Research, Vol. 16, No. 1, 1978, pp. 27-44. 

Lee, K., and Sills, G. C, "A Moving Boundary Approach to Large Strain 
Consolidation of a Thin Soil Layer," Proc. 3rd International 
Conference on Numerical Methods in Geomechanics . W. Wittke, Editor, 
Rotterdam, 1979, pp. 163-173. 

Lee, K., and Sills, G. C, "The Consolidation of a Soil Stratum, 
Including Self-weight Effects and Large Strains," Numerical and 
Analytical Methods in Geomechanics . Vol. 5, 1981, pp. 105-428. 

Lee, P. T,, "Deposition and Scour of Clay Particles Due to Currents," 
M.S. Thesis , Asian Institute of Technology, Bangkok, Thailand, 1974. 

Leendertse, J. J., Alexander, R. C, and Liu, S. K., "Three-Dimensional 
Model for Estuaries and Coastal Seas: Vol. I, Principles of 
Computation," Report R-1417-0WRR, Rand Corporation, Santa Monica, 
California, December, 1973. 

Leimkuhler, W., Connor, J., Wang, J., Christodoulou, G., and Sundgren, 
A., "Two-Dimensional Finite Element Dispersion Model," Symposium on 
Modeling Techniques . ASCE, San Francisco, California, September, 
1975, pp. 1467-1486. 

Lerman, A., and Weiler, R. R. "Diffusion and Accumulation of Chloride 
and Sodium in Lake Ontario Sediment," Earth and Planatary Sci ence 
Letters . Vol. 10, 1970. pp. 150-156. 

Li, Y. H., and Gregory, S., "Diffusion of Ions in Sea Water and in Deep- 
Sea Sediments," Geochimica et Cosmoch imica Acta, Vol. 38, 1974, 
pp. 703-714. ~ 

Liu, H., and Cheng, A. H. D., "Modified Fickian Model for Predicting 
Dispersion," Journal of the Hydraulics Division. ASCE, Vol. 106, 
No. HY6, June, 1980, pp. 1021-1040. 

Liu. S. K., and Leendertse, J., "Multi-dimensional Numerical Modeling of 
Estuaries and Coastal Seas," Advances in Hydroscience , Vol. 11, 
Academic Press, New York, 1978^ 

MacArthur, R. C, "Turbulent Mixing Processes in a Partially Mixed 
Estuary," Ph.D. Dissertation. University of California, Davis, 
California, 1979. 



Manheim, F. T., "The Diffusion of Ions in Unconsolidated Sediments," 
Earth and Planatary Science Letters. Vol. 9, 1970, pp. 307-309. 



342 



Mehta, A. J., "Depositional Behavior of Cohesive Sediments," Ph.D. 
Dissertation . University of Florida, Gainesville, Florida, 1973. 

Mehta, A. J., "Review of Erosion Function for Cohesive Sediment Beds," 
Proc. First Indian Conference on Ocean Engineering. Indian Institute 
of Technology, Madras, India, Vol. 1, February, 1981, pp. 122-130. 

Mehta, A. J., and Partheniades, E., "Depositional Behavior of Cohesive 
Sediments," Technical Report No. 16. Coastal and Oceanographic 
Engineering Laboratory, University of Florida, Gainesville, Florida. 
March, 1973. 

Mehta, A. J., and Partheniades, E., "An Investigation of the 

Depositional Properties of Flocculated Fine Sediments," Journal of 
Hydraulics Research . International Association for Hydraulics 
Research, Vol. 13, No. 14, 1975, pp. 361-381. 

Mehta, A. J., and Partheniades, E., "Kaolinite Resuspension Properties," 
Journal of the Hydraulics Division. Proc. ASCE, Vol. 105, No. HY4, 
March, 1979, pp. 411-416. 

Mehta, A. J., and Hayter, E. J., "Preliminary Investigation of Fine 
Sediment Dynamics of Cumbarjua Canal, Goa, India," UFL/COEL-81-012. 
Coastal and Oceanographic Engineering Department, University of 
Florida, Gainesville, Florida, December, 1981. 

Mehta, A. J., Parchure, T. M., Dixit, J. G., and Ariathurai, R., 
'Resuspension Potential of Deposited Cohesive Sediment Beds," In 
Estuarine Comparisons . V. S. Kennedy, Editor, Academic Press, New 
York, 1982a, pp. 591-609. 

Mehta, A. J., Partheniades, E., Dixit, J., and McAnally, W. H., 
"Properties of Deposited Kaolinite in a Long Flume," Proc. of 
Hydraulics Division Conference on: Applying Research to Hydraulic 
Practice , ASCE, Jackson, Mississippi, August, 1982b. 

Migniot, P. C, "A Study of the Physical Properties of Different Very 
Fine Sediments and Their Behavior Under Hydrodynamic Action," La 
Houille Blanche. No. 7, 1968, pp. 591-620. (In French, with English 
abstractK 

Mitchell, J. K., "Fundamental Aspects of Thixotropy in Soils," 
Transactions of the ASCE. Vol. 126, Pt. 1, 1961, pp. 1586-1620. 

Mitchell, J. K., Fundamentals of Soil Behavior, John Wiley & Sons. New 
York, 1976. 

Mitchell, J. K., Singh, A., and Campanella, R. G., "Bonding Effective 
Stresses and Strength of Soils," Journal of the Soil Mechanics and 
Foundation Division. ASCE, Vol. 95, Ho. SMS, September, 1969. 
pp. 1219-1246. 



343 



Murray, S» P., and Siripong, A., "Role of Lateral Gradients and 

Longitudinal Dispersion in the Salt Balance of a Shallow, Well -Mixed 
Estuary," In Estuarine Transport Processes . B. Kjerfve, Editor, 
University of South Carolina Press, Columbia, South Carolina, 1978, 
pp. 113-124. 

Nichols, M., Faas, R., and Thompson, G., "Estuarine Fluid Mud: Its 
Behavior and Accumulation," Final Report, Virginia Institute of 
Marine Science, Gloucester Point, Virginia, April, 1979. 

Norton, W.R., King, I. P., and Orlob, G.T., "A Finite Element Model for 
Lower Granite Reservoir," Walla Walla District U.S. Army Report , 
prepared by Water Resources Engineers, Inc., March, 1973. 

Nriagu, J. 0., "Dissolved Silica in Pore Waters of Lakes Ontario, Erie, 
and Superior Sediments," Limnology and Oceanography. Vol. 23, No. 1, 
January, 1978, pp. 53-67. 

O'Connor, B. A., "Siltation in Dredged Channels," In First International 
Symposium on Dredging Technology, S. K. Hemmings, Editor, Canterbury, 
United Kingdom, 1975. 

O'Connor, B.^^A., and Zein, S., "Numerical Modelling of Suspended 
Sediment," Proc. Fourteenth Coastal Engineering Conference, ASCE, 
Copenhagen, Denmark, June, 1974, pp. 1109-1128. 

Odd, N. V. M., and Owen, M. W., "A Two-Layer Model for Mud Transport in 
the Thames Estuary," Proc. of the Institution of Civil Engineers , 
London, Supplement (ix), 1972, pp. 175-205. 

Officer, C. B., Physical Oceanography of Estuaries . John Wiley, New 
York, 1976. 

Officer, C. B., "Physical Dynamics of Estuarine Suspended Sediments," 

Marine Geology. Vol. 40, 1981, pp. 1-14. 

Okubo, A., "Effect of Shoreline Irregularities on Streamwise Dispersion 
in Estuaries and Other Embayments," Netherlands Journal of Sea 
Research , Vol. 6, 1973, pp. 213-224. 

O'Melia, C. R., in Physicochemical Processes for Water Quality Control . 
W. J. Weber, Editor, Wiley-Interscience, New York, 1972, pp. 61-109. 

Onishi, Y., and Wise, S. E.. "Finite Element Model for Sedimentation and 
Toxic Contaminant Transport in Streams," Proc. Speciality Conference 
on Conservation and Utilization of Water and Energy Resources , ASCE, 
San Francisco, California, August, 1979, pp. 144-150. 

Orlob, G. T., Shubinski, R. P., and Feigner, K. D., "Mathematical 
Modeling of Water Quality in Estuarial Systems." ASCE National 
Symposium on Estuarine Pollution . Stanford University, California, 
August, 1967. 



344 



Owen, M. W., "A Detailed Study of the Settlinq Velocities of an Estuary 
Report No. INT 78 . Hydraulics Research Station, Wallinqford 
United Kingdom, September, 1970. 

Owen, M. W., "The Effect of Turbulence on the Settling Velocities of 
Silt Floes," Proc. of Fourteenth Congress of I.A.H.R., Vol. 4, Paris 
August, 1971, pp. 27-32. ~ 

Owen, M. W. , "Erosion of Avonmouth Mud," Report No. INT 150 . Hydraulics 
Research Station, Wallingford, United Kingdom, September, 1975. 

Owen. M. W., "Problems in the Modeling of Transport, Erosion, and 
Deposition of Cohesive Sediments," In The Sea VI. Marine Model ing 
E. D. Goldberg, I. N. McCave, J. J. O'Brien and J. H. Steel, Editors, 
Wiley-Interscience, New York, 1977. pp. 515-537. 

Paaswell, R. E., "Causes and Mechanisms of Cohesive Soil Erosion- The 
State of the Art," Special Report. 135. Highway Research Board, 
Washington, D.C., 1973, pp. 52-74. 

Parchure, T. M., "Effect of Bed Shear Stress on the Erosional 

Characteristics of Kaolinite," M.S. Thesis , University of Florida 
Gainesville, Florida, December, 1980. 

Parchure, T. M., "Erosional Behavior of Deposited Cohesive Sediments," 
Ph.D. Diss ertation. University of Florida, Gainesville, Florida, 
1983 . 

Parker W. R., "Sediment Dynamics," Unpublished Lecture Notes . University 
of Florida, Gainesville, Florida, February, 1980. 

Parker, W. R., and Kirby, R., "Fine Sediment Studies Relevant to 

Dredging Practice and Control," Proc. Second International Sympos ium 

on Dredging Technology . BHRA, Paper B2, Texas A & M University. 

College Station. November, 1977. 

Parker, W. R., and Kirby, R., "Time Dependent Properties of Cohesive 
Sediment Relevant to Sedimentation Management-European Experience," 
In Estuarine Comparisons , V. S. Kennedy, Editor, Academic Press 
New York, 1982. * 

Parker, W. R., and Lee, K., "The Behavior of Fine Sediment Relevant to 
the Dispersal of Pollutants," ICES Workshop on Sediment and Pollutan t 
Interchang e in Shallow Seas . Texel , United Kingdom, September, 1979. 

Parker, W. R., Sills, G. C, and Paske, R. E. A., "In -situ Bulk Density 
Measurement in Dredging Practice and Control," In FTrst" International 
Symposium on Dredging Technology. S. K. Hemmings, Editor, Canterbury. 
United Kingdom, 1975. 

Partheniades, E., "A Study of Erosion and Deposition of Cohesive Soils 
in Salt Water," Ph.D. Dissertation. University of California. 
Berkeley, California, 1962. 



345 



Partheniades, E., "A Summary of the Present Knowledge on the Behavior of 
Fine Sediments in Estuaries," Technical Note No. 8, Hydrodynamics 
Lab, M.I.T., Cambridge, Massachusetts, 1964. 

Partheniades, E., "Erosion and Deposition of Cohesive Soils," Journal of 
the Hydraulic Division , ASCE, Vol. 91, No, HYl, January, 1965^ 
pp. 105-138. 

Partheniades, E., "Erosion and Deposition of Cohesive Materials," in 
River Mechanics, Vol. II, H. W. Shen, Editor, Fort Collins, Colorado. 
1971. 

Partheniades, E., "Unified View of Wash Load and Bed Material Load," 
Journal of the Hydraulics Division. ASCE, Vol. 103, No. HY9, 1977. 
pp. 1037-1057. 

Partheniades, E., Kennedy, J. R., Etter, R. J., and Hoyer, R. P., 
"Investigations of the Depositional Behavior of Fine Cohesive 
Sediments in an Annular Rotating Channel," Report No. 96 , 
Hydrodynamics Lab., M.I.T., Cambridge, Massachusetts, 1966. 

Partheniades, E., Cross, R. H., and Ayoro, A., "Further Results on the 
Deposition of Cohesive Sediments," Proc. Eleventh Conference on 
Coastal Engineering , ASCE, Ch. 47, 1968, pp. 723-742. 

Peterson, J. P., Castro, W. E., Zielinski, P. B., and Beckwith, W. F., 
"Enhanced Dispersion in Drag Reducing Open Channel Flow," Journal of 
the Hydraulics Division. ASCE, Vol. 100, No. HY6, June, 197T; 
pp. 773-785. 

Postma, H., "Sediment Transport and Sedimentation in the Estuarine 

Environment," in Estuaries . G. H. Lauff, Editor, American Association 
for Advancement of Science Publication 83, Washington, D.C., 1967, 
pp. 158-179. 

Preston, A., Jefferies, D. F., Dutton, J. W. R., Harvey, B. R., and 
Steele, A. K., "British Isles Coastal Waters: The Concentration of 
Selected Heavy Metals in Sea Water, Suspended Matter and Biological 
Indicators - a Pilot Survey," Journal of Environ mental Pollution, 
Vol. 3, 1972, pp. 69-72. ~~ 

Pritchard, D., "Dispersion and Flushing of Pollutants in Estuaries," 
Journal of the Hydraulics Division. ASCE, Vol. 95, No. HYl, January, 
1969, pp. 115-124. 

Quirk, J. P. and Schofield, R. K., "The Effect of Electrolyte 

Concentration on Soil Permeability," Journ al of Soil Science, Vol. 6. 
No. 2, 1955. 

Rao, L. V. G., Cherian, T. , Varma, K. K., and Vardachari, V. V. R., 

"Hydrographic Conditions and Flow Pattern in Cumbarjua Canal, Goa," 

Indian Journal of Marine Sciences . Vol. 5, December, 1976, 
pp. 163-168. 



346 



Richards, A. F., Hirst, T. J., and Parks, J. M., "Bulk Density - Water 
Content Relationship in Marine Silts and Clays," Journal of 
Sedimentary Petrology , Vol. 44, No. 4, 1974, pp. 1004-1009. 

Rosillon, R., and Volkenborn, C, "Sedimentacion de Material Cohesivo en 
Agua Salada," Thesis , University of Zulia, Maracaibo, Venezuela, 
November, 1964. 

Sargunam, A., Riley, P., Arulanandan, K., and Krone, R.B., "Effect of 
Physico-chemical Factors of the Erosion of Cohesive Soils," Journal 
of the Hydraulics Division , ASCE, Vol. 99, No. HY3, March, 1973, 
pp. 553-558. 

Sayre, W. W., "Dispersion of Mass in Open Channel Flow," Hydraulics 
Papers, No, 3, Colorado State University, Fort Collins, Colorado, 
February, 1968. 

Sayre, W. W., "Dispersion of Salt Particles in Open Channel Flow," 
Journal of the Hydraulics Division, ASCE, Vol. 95, No. HY3, May, 
1969, pp. 1009-1038. 

Schiffman, R. L., and Cargill, R. K. W., "Finite Strain of Sedimenting 
Clay Deposits," Proc. Tenth International Conference on Soil 
Mechanics and Foundation Engineering , Stockholm, Vol. 1, 1981, 
pp. 239-242. 

Sherard, J. L., Decker, R. S., and Ryka, N. L., "Piping in Earth Dams of 
Dispersive Clay," ASCE Soil Mechanics and Foundation Conference , 
Purdue University, Lafayette, Indiana, June, 1972. 

Smith, R., "Coriolis, Curvature and Buoyancy Effects Upon Dispersion in 
a Narrow Channel," In Hydrodynamics of Estuaries and Fjords , 
J. C. J. Nihaul, Editor, Elsevier, Amsterdam, 1978, pp. 217-232. 

Spangler, M. G., and Handy, R. L., Soil Engineering , Harper and Row, 
New York, 1982. 

Srivastava, M., "Sediment Deposition in a Coastal Marina," UFL/COEL/MP- 
83/1, Coastal and Oceanographic Engineering Department, University of 
Florida, Gainesville, Florida, 1983. 

Sumer, M., "Turbulent Dispersion of Suspended Matters in a Broad Open 
Channel," Proc. International Association for Hydraulic Research 
Conference . Paris, 1971. ~ 

Sumer, S. M., and Fischer, H. B., "Transverse Mixing in Partially 

Stratified Flow," Journal of the Hydraulics Division. ASCE, Vol. 103, 
No. HY6. June, 1974, pp. 587-600. 

Taylor. D. W., Fundamentals of Soil Mechanics. John Wiley and Sons, 
New York, 1948: ' ~ 

Taylor, G. I.. "Dispersion of Soluble Matter in Solvent Flowing Slowly 
Through a Tube," Proc. Royal Society of London . Series A, Vol. 219, 
1953, pp. 186-203. 



347 

Taylor, G, I., "The Dispersion of Matter in Turbulent Flow Through a 
Pipe," Proc. Royal Society of London , Series A, Vol. 223, May, 1954, 
pp. 446-468. 

Taylor, R, B., "Dispersive Mass Transport in Oscillatory and 

Unidirectional Flows," Ph.D. Dissertation. University of Florida, 
Gainesville, Florida, 1974. 

Teeter, A, M., "Investigations on Atchafalaya Bay Sediments," Proc. 
Conference on Frontiers in Hydraulic Engineering, ASCE, Cambridge, 
Massachusetts, August, 1983, pp. 85-90. 

Terzaghi, K., "Die Theorie der Hydrodynamischen Spanungserscheinungen 

und ihr Erdbautechnisches Anwendungsgebeit," Proc. First 

International Congress of Applied Mechanics , Delft, The Netherlands, 
No. 1, 1924, pp. 228-294. 

Terzaghi, K., and Peck, R. B., Soil Mechanics in Engineering Practice . 
John Wiley and Sons, New York, 1960. 

Thorn, M. F. C, "Physical Processes of Siltation in Tidal Channels," 
Proc. Hydraulic Modelling Applied to Maritime Engineering Problems, 
ICE, London, 1981, pp. 47-55. 

Thorn, M. F. C, and Parsons, J. G., "Properties of Grangemouth Mud," 
Report No. EX781, Hydraulics Research Station, Wallingford, United 
Kingdom, July, 1977. 

Thorn, M. F. C., and Parsons, J. G., "Erosion of Cohesive Sediments in 
Estuaries: An Engineering Guide," Proc. Third International 
Symposium on Dredging Technology . Paper Fl, March, 1980. 

van Olphen, H., An Introduction to Clay Colloid Chemistry . Interscience 
Publishers, New York, 1963. 

van Rijn, L. C. , "Pump-Filter Sampler, Design of an Instrument for 
Measuring Suspended Sand Concentrations in Tidal Conditions," 
Research Report S 404-1 . Delft Hydraulics Laboratory, The 
Netherlands, June, 1979. 

Wang, S., "Friction in Hurricane - Induced Flooding," Ph.D. 

Dissertation. University of Florida, Gainesville, Florida, 1983. 

Ward, P. R. B., "Transverse Dispersion in Oscillatory Channel Flow," 
Journal of the Hydraulics Division. ASCE, Vol. 100, No. HY6, June, 
1974, pp. 755-772. 

Ward, P. R. B., "Measurements of Estuary Dispersion Coefficient," 
Journal of the Environmental Engineering Division. ASCE, Vol. 102, 
No. EE4, August, 1976, pp. 885-859. 

Weckmann, J., "Sediment Management of a Coastal Marina," M.E. Thesis , 
University of Florida, Gainesville, Florida, 1979. 



348 



Whitehouse, U. G., Jeffrey, L. M., and Debbrecht, J. D., "Differential 
Settling Tendencies of Clay Minerals in Saline Waters," Proc. Seventh 
National Conference on Clays and Clay Minerals , 1960, pp. 1-79. 

Whitmarsh, R. B., "Precise Sedinent Density by Gamma-Ray Attenuation 
Alone," Journal of Sedimentary Petrology . Vol. 41, 1971, pp. 882-883. 

Wilson, W., and Bradley, D., "Specific Volume of Sea Water as a Function 
of Temperature, Pressure and Salinity," Deep-Sea R esearch, Vol. 15, 
1968, pp. 355-363. 

Yeh, H. Y., "Resuspension Properties of Flow Deposited Cohesive Sediment 
Beds," M.S. Thesis. University of Florida, Gainesville, Florida. 
1979. 

Zienkiewicz, 0. C, The Finite Element Method in Engineering Scien ce. 
McGraw-Hill, London, 1977. 

Zimmerman, J. T. F., "Dispersion by Tide-Induced Residual Current 

Vortices," In Hydrodynamics of Estuaries and Fjords . J. C. J. Nihoul, 
Editor, Elsevier, Amsterdam, 1978, pp. 207-216. 

Znidarcic., D., "Laboratory Determination of Consolidation Properties for 
Cohesive Soil," Ph.D. Dissertation . University of Colorado, Boulder, 
Colorado, 1982. 



BIOGRAPHICAL SKETCH 



Earl Joseph Hayter was born November 19, 1954, in Coral Gables, 
Florida. At the age of 9 his family moved to Apopka, Florida, where he 
completed his primary and secondary education and graduated from Lyman 
High School in 1972. He attended Florida Technological University from 
September 1972 to June 1974. In September 1974 he transferred to 
Florida Institute of Technology and majored in physical oceanography. 
He graduated from F.I.T. in June 1975 with a Bachelor of Science degree 
in oceanography. In September 1976 he enrolled as a graduate student in 
the Coastal and Oceanographic Engineering Department at the University 
of Florida. He graduated from the University of Florida in March 1979 
with a Master of Science degree in coastal and oceanography 
engineering. From September 1977 to May 1979 he worked for Suboceanic 
Consultants, Inc., in Naples, Florida. 

He reentered graduate school at the University of Florida in June 
1979 in the Department of Civil Engineering to work toward the Doctor of 
Philosophy degree. On June 21, 1980, he married Janet Griffen. He 
worked as a graduate research assistant in the Coastal and Oceanographic 
Engineering Department during the periods September 1976 - September 
1977 and June 1979 - September 1981. From October 1981 to September 
1983, he was supported by the Water Resources Division of the U.S. 
Geological Survey under their thesis support program. 

349 



I certify that I have read this study and that in my 
opinion 1t conforms to acceptable standards of scholarly 
presentation and is fully adequate in scope and quality, as 
a dissertation for the degree of Doctor of Philosophy. 

Bent A. Christensen, C hai rman 
Professor of Civil Engineering 



I certify that I have read this study and that in my 
opinion it conforms to acceptable standards of scholarly 
presentation and is fully adequate in scope and quality, as 
a dissertation for the degree of Doctor of Philosophy. 




/A sh^*W^J. Mehta, C o"- C h a i r ma h 
Associate Professor of Coastal 
and Oceanographic Engineering 



I certify that I have read this study and that in my 
opinion it conforms to acceptable standards of scholarly 
presentation and is fully adequate in scope and quality, as 
a dissertation for the degree of Doctor of Philosophy. 




Barry ^ Benedict 

Professor of Civil Engineering 



I certify that I have read this study and that in my 
opinion it conforms to acceptable standards of scholarly 
presentation and is fully adequate in scope and quality, as 
a dissertation for the degree of Doctor of Philosophy. 





WAjL^ ex rQ.r^!l^^4^ 

/J^mes ~. Eades 

Associate Professor of Geology 



Tin's dissertation v/as submitted to the Dean of the College of 
Engineering and to the Graduate Council, and was accepted as partial 
fulfillment of the requirements for the degree of Doctor of Philosophy. 



December, 1983 



Dean, College of Engineering 



Dean, Graduate School